Linear and Multilinear Algebra, 1990, Vol. 28, pp. 3-33 Reprints available directly from the publisher Photocopying permitted by license only C9 199 Gordon and Breach Science Publishers S.A, Printed in the United States of America The Largest Eigenvalue of a Graph: A Survey i\ ~ D. CVETKOVIC Department of Mathematics, Faculty of Electrical Engineering, University of Belgrade, PO Box 816, 11001 Beograd, Yugoslavia

P. ROWLINSON Department of Mathematics, University of Stirling, Stirling FK9 4LA, Scotland

(Received March 1, 1990)

Ths article is a survey of results concerning the largest eigenvalue (or index) of a graph, categorized as follows: (i) inequalities for the index, (2) graphs with bounded index, (3) ordering graphs by their indices, (4) graph operations and modifications, (5) random graphs, (6) applications,

INTRODUCTION Almost all results related to the theory of graph spectra and published before 1984 are summarized in the monographs (27) and (28). In view of the rapid growth of the subject in subsequent years it is no longer reasonable to expect a single book to provide a comprehensive survey of the latest results. Instead it seems more ap- propriate that expository articles should be devoted to specific topics. For example the paper (68) reflects the recent realization that many results from analytic proba- bility theory have implications for the spectra of infinite graphs. Here we survey what is known about the largest eigenvalue of a finite graph. This topic embraces early results which go back to the very beginnings of the theory of graph spectra, together with recent developments concerning ordering and pertur- bations of graphs. Proofs which appear in (27) and (28) are not repeated here. We discuss only finite undirected graphs without loops or multiple edges, and we start with some basic definitions. Let G be a graph with n vertices, and let A be a (0,1)- adjacency matrix of G, regarded as a matrix with real entries. Since A is symmetric, its eigenvalues Ài, Ài,. .., Àn are real, and we assume that Ài ~ Ài ~ .. . ~ Àn. These eigenvalues are independent of the ordering of the vertices of G, and accordingly we write Ài(G) = Ài(A) = Ài (i = i,...,n) and refer to Ài,....,ÀIi as the spectrum of G. The largest eigenvalue Ài is called the index of G (orl spectral radius of A). We call det(xI - A) the characteristic polynomial of G, denoted by ltG(x). The distinct eigenvalues of G wil be denoted by lli,..., llm, ordered as required. Since A is a non-negative matrix, some general information on the spectrum of G is provided by the Perron-Frobenius theory of matrices (45, 49, 57, 65, 67). In particular, if G is connected then A is irreducible and so there exists a unique positive unit eigen- vector corresponding to the index Ài. This vector we call the principal eigenvector of G: note that entries corresponding to vertices in the same orbit of Aut( G) are

3 LARGEST EIGENVALUE OF A GRAPH 5 4 D. CVETKOVlC AND P. ROWLINSON

Let A denote the adjacency matrix of an n-vertex graph G. For any non-zero equal. We shall also use implicitly the fact that if G' is obtained from G by adding vector vE W the Rayleigh quotient vTAv/vTv is a lower bound for À1(G), as can an edge then À1(G') ~ À1(G), with strict inequality when G is connected: this is an be seen by diagonalizing A. If we take v to be the all-1 vector j then we obtain the immediate consequence of the fact that the spectral radius of a non-negative matrix inequality d ~ À1 of Theorem 1.1. Similarly, IAvl/lvl is a lower bound for À1, and increases with each entry. Further fundamental results in matrix theory which serve the next two results may be obtained by setting v = j, v = ei respectively, where the as a background to problems concerning the largest eigenvalue may be found in , i-th vertex of G has maxmal degree and ei = (6ii,...,6inl. (19, 35). We note in passing that although the eigenvalues of a directed multigraph need not be real, such a digraph has a positive eigenvalue À1 such that IÀI ~ À1 for THEOREM 1.2 (Hofmeister (54)) If the vertices of G have degrees d1,d2,.. .,dn then all eigenvalues À. Some results on the spectral radius of digraphs may be found in À1(G) ~ l(lln) ,,7=1 dr. (6, 83, 93, 98). For differing approaches to the spectra of infinite graphs, with some results on the largest eigenvalue, the reader is referred to Chapter VI of the mono- THEOREM 1. (Nosal (70); Lovász & Pelikán (62)) If dmax is the maxmal degree of graph (27), the expository article (68) and the paper (17). For the index in particular, a vertex in G then À1(G) ~ ldmax. see (8). The paper (96) extends to infinite graphs a result on finite graphs whose in- dex does not exceed V2 + v' (Theorem 2.4). Some discussion of the index of a Hofmeister went on to show that for any graph G, there exists a real number p, graph may be found in the expository papers (84, 91, 101). unique if G is non-regular, such that À1 (G) = \! (11 n) ,,7=1 dl. Finally we point out that several of the authors' results mentioned in this article Theorems 1.1 and 1.3 show that the index of a graph is controlled by the maxmal were conjectured on the basis of numerical evidence furnished by the expert system degree in the sense that À1 is bounded above and below by a function of dmax. Hoff- "Graph" (22, 23, 31, 90). In some instances, proofs were completed by using the man (51) observes that Ramsey-type arguments may be used to prove that À1(G) is system to check outstanding cases. controlled by the least number t such that neither Kt nor the star K1,t is an induced subgraph of G. We state one more result which demonstrates the interplay between index and 1. BOUNDS FOR THE INDEX OF A GRAPH vertex degrees. In a regular graph G of degree r on n vertices the number Nk Here we give upper and lower bounds for the index of a graph which are ex- of walks of length k is given by Nk = nrk; thus N Nk I n = r and this suggests that pressed in terms of various graph invariants. These bounds are interpreted and used N Nk I n in the general case might be regarded as a certain kind of mean value from different viewpoints in other sections of this article: in Section 2 for example of vertex degrees. Accordingly d = limk~+oo NNkln is defined to be the dynamic we survey classes of graphs defined by some bounds on the index. The discussion mean of the vertex degrees of an arbitrary graph G. of graph ordering in Section 3 and of graph perturbations in Section 4 is often con- the vertex cerned with bounds on the index. In §6.1 we return to the question of bounding THEOREM 1.4 (D. Cvetkovic (20)) For any graph G, the dynamic m~an of other graph invariants in terms of the index. degrees is equal to the index of G. In the fundamental paper (18) on the theory of graph spectra, Collatz and Sino- We give next some inequalities for the index of a graph involving the number n gowitz observed that the index À1 of a connected graph on n vertices satisfies the of vertices and the number e of edges. A part of Theorem 1.1 may be stated in inequality the form À1 ~ 2e I n, since 2e I n is the mean degree. An upper bound in terms of e 11 2cos-n+1 .: À1'':- - n - 1. and n may be obtained by maxmizing À1 subject only to the constraints ,,7=1 Ài = o and ,,7=1 ÀT = 2e (that is, tr(A) = 0 and tr(A2) = ,,7=1 di). Then we obtain the the The lower bound is attained by the path Pn while the upper bound is attained by following result of Wilf. complete graph Kn. If we omit the assumption of connectedness, then for a graph without edges we have À1 = 0 and otherwise À1 ~ 1. THEOREM 1.5 (100) For any graph G, À1(G) ~ V2e(1-1In). A reformulation of inequalities from the theory of non-negative matrices (67, The inequality in the next result was announced by Schwenk in (82), and a proof Chapter 2) yields the following theorem. by Yuan appeared some thirteen years later. THEOREM 1.1 Let dmim d, dmax respectively be the minimal, mean and maxmal val- THEOREM 1.6 (103) For any connected graph G, À1(G) ~ v2e - n + 1. Equality ues of the vertex degrees in a connected graph G. If À1 is the index of G then holds if and only if G is the star K1,n-1 or the complete graph Kn. dmin ~ d ~ À1 ~ dma. Proof Suppose that G has vertices 1,2,..., n and adjacency matrix A. Let di de- note the degree of vertex i and let Ci be the i-th column of A. Let x be the principal Equality in one place implies equality throughout; and this occurs if and only if G is eigenvector of A, say x = (xi,...,Xn)T, and let Vi be the vector whose j-th entry is reguar. LARGEST EIGENVALUE OF A GRAPH 7 6 D. CVETKOviC AND P. ROWLINSON

A lower bound for the chromatic number is given in the next theorem, the proof Xj if j '" i ("j is adjacent to i") and 0 otherwise. We have c7V¡ = c7x = ),1Xi, and so of which is outlned in (28, Chapter 3). by the Cauchy-Schwarz inequality, THEOREM 1.8 (A. J. Hoffman (52)) Let),1 and ),n be the largest and the least eigen- value of a graph G. Then the chromatic number ¡(G) satisfies ),r xl ~ lei IZlv¡\Z = di (1 - L xi) . j-fi ),1 ¡(G) 2' 1 + I),n I. Summing over i, we obtain Subsequently D. Cvetkovic (21) proved that ),r ~ 2e - tdi (LXY) . n i=1 j-fi ¡(G) 2' n - ),1' (1.1) But a result which was rediscovered in (38). Let ~(G) be the size of the largest clique in G: ~(G) is called the clique number of G. Since ¡(G) ~ ~(G) one might ask whether nj(n - ),1) is also a lower bound t.d¡ (j;xi) ~ t.diX¡ + t.d¡ (1;/1) for ~(G). This was proved in (39) for planar graphs, while the authors of (38) offer n n ( ) n the inequality -1 n ~LdiXr+L L xy =LLxy=n-1 ~(G):?-+-. (1.2) i=1 i=1 i'f-fi i=1 di - 3 n - ),1 Both (1.1) and (1.2) can be improved however because Wilf has shown: and so ),t ~ 2e - n + 1. Equality holds if and only if for each i, either di = 1 or di = n - 1, and the result follows. · THEOREM 1.9 (Wilf (102)) For any graph G, n There are several inequalities involving the index ),1 and the chromatic number ~(G):?-. . - n - ),1 ¡( G) of a graph.

THEOREM 1.7 (H. S. Wilf (100)) For the chromatic number ¡(G) of a graph G we In addition he was able to derive a better lower bound for K,( G) which involves have ),1 and a corresponding eigenvector. ¡(G) ~ 1 + ),1' The following condition for K,( G) ~ 3 is established in (28, p. 86): For a connected graph G, equality holds if and only if G is complete or a cycle of odd THEOREM 1.10 (Nosal (70)) Let),1 be the index and e the number of edges of a length. graph G. If ),1 ? ve, then G contains a triangle. Proof (101) Delete edges from G until a critical graph G is obtained: thus the This result is extended in (9) to provide conditions on ),1 which ensure a girth deletion of any further edge would reduce the chromatic number. In G all vertex no greater than 2k + 1 for k a positive integer. Also a condition is presented which degrees are at least ¡(G) - 1. If dmin(G) denotes the minimal degree of G then we guarantees that the girth is no more than four. have Finally we mention two results involving the index of the complement G of a ¡(G) -1 ~ dmin(G) ~ ),1(G) ~ ),1(G). graph G. The first may be proved by applying Theorems 1.1 and 1.5 to G and G. If G is connected and equality holds then G is regular of degree ¡(G) - 1. The THEOREM 1.11 (E. Nosal (70), A. T. Amin and S. L. Hai6~i (1)) Let G be a graph second assertion of the Theorem now follows from the classical result of Brooks on n vertices. We have (10), which for a connected graph G states that ¡(G) ~ 1 + dmax, with equality if and only if G is complete or a cycle of odd length. · n -1 ~ ),1(G) + ),1(G) ~ Vi(n -1). Analogous results for point-arboricity and related invariants (61) have been ob- tained by Lick (60): see (28, pp. 90-91). A computational comparison of several THEOREM 1.12 (D. C. Fisher (40)) Let fez) = 1- C1Z + czzz - C3Z3 + ... where Ck bounds for the chromatic number of a graph appears in (37): the spectral bound is the number of complete subgraphs on k vertices in G. If r (G) is the reciprocal of from Theorem 1.7 is reported to be in the middle of the list. the smallest real root of fez) then ),1(G) ~ reG) - 1. LARGEST EIGENVALUE OF A GRAPH 9 8 D. CVETKOVIC AND P. ROWLINSON

Proof We give only a brief outline of the proof. Let V(G) denote the set of vertices of G, and let M (G) be the monoid generated by V (G) subject to the re- lations uv = vu precisely when uv is an edge of G. For n = 0 let Pn be the num- ber of n-Ietter words in M(G). An inclusion-exclusion argument (41) shows that for n)o 0, Pn = cipn-i - CZPn-Z +... + (-It+lcnPo, with po = 1. Thus f(z)(po +

piz + P2ZZ + ...) = 1, and it follows that limn--oop~/n = reG). Next let qn be the .n ,"..:::,-~: number of n-Ietter words w on V(G) as alphabet, subject to the restriction that adjacency in G precludes adjacency in w. We have Pn;: qn = jT(A + I)j, where A \ I - i/n i/n i/n - "- denotes the adjacency matrix of G. Hence pn ;: qn , and qn -+ Ài(G) + 1 as / ,--- ./ n -+ 00. The result follows. · en In edges'

2. GRAPHS WITH BOUNDED INDEX Some interesting classes of graphs can be obtained by prescribing an upper bound for the index. The graphs for which .\i :: 2 can be determined essentially because their vertices have mean degree:: 2 and maxmum degree:: 4 (cf. Theorems 1.1 i and 1.3): the classification of such graphs is given in Section 2.1. In Section 2.2 we G2 discuss the graphs for which ..i :: ";2 + 0: note that if 7 denotes the golden ratio

'ij lci + 0) then ";2 + 0 = 73/2 = 7i/2 + 7-i/2 ~ 2.058171. The significance of this ~\1!; 3 ~! number as an upper bound is explained (in part) as follows. If G is a connected "" mi graph which is neither a tree nor a cycle then for some m ?: 3 it has a subgraph Hm ¡ID- consisting of an m-cyc1e and a pendant edge. It follows from a result of Hoffman 4 5 6 4 2 f (50) that Ài(Hm) approaches ";2 +.. from above as m -+ 00, and consequently 1i!: G) il1 ..i(G))o ";2 + ... Thus apart from cycles, the connected graphs with index at most ";2 + 0 are trees. Theorem 2.4 provides a classification of the trees G for which FIGURE 1. I," li 20( ..i(G):: ";2 + ... Since ";2 +.. = limm--oo..i(Hm), the number ";2 +.. is f~ . said to be a limit point of graph indices: such limits are the topic of Section 2.3, ;¡,:, m where ";2 + .. wil assume further significance. then either G = G2 or a second path has length less than 3. In the latter case G is ~. an induced subgraph of G3 or some Wn. Finally, if the maxmal degree of a vertex '1l~~ in G is 2 then G is a path and hence an induced subgraph of some en' . ')1 2.1. Graphs whose largest eigenvalue does not exceed 2 00:;.1', ~!~j We start with the following result. ~j Theorem 2.1 and its proof are due to J. H. Smith (91), and accordingly graphs

Wl, : THEOREM 2.1 (91) The connected graphs whose largest eigenvalue does not exceed 2 from Figure 1 are often called Smith graphs in the literature (see, for example, (29)). are precisely the induced subgraphs of the graphs shown in Figure 1. Seidel (84) proposed the name 'Coxeter graphs' because the graphs inqueestion ap- pear implicitly in Coxeter's work on discrete groups generated by reflections in hy- Note In Figure 1, Wo = Ki,4 and each graph has index 2: the numbers attached perplanes. Since the topic of this article is a part of graph theory rather than group to vertices are components of a corresponding (positive) eigenvector. theory, we prefer the elegant graph-theoretic proof by Smith. For another proof Proof Let G be a connected graph with ..i(G):: 2. Since ..i increases strictly of Theorem 2.1 see (69). For the rôle of these graphs in algebra see, for example, monotonically with the addition of vertices, provided the graph remains connected, (47), where they appear as Dynkin diagrams. The Smith graphs are very important for another part of the theory of graph spectra, namely the study of graphs with G is either a cycle en or a tree; moreover Wo is the only possible tree with a vertex of degree greater than 3. If the maxmal degree is 3, then either G = Wn for some least eigenvalue bounded below by -2. These graphs have been characterized by n )0 0 or G has a unique vertex of degree 3 with three paths attached. In the second Cameron, Goethals, Seidel and Shult in terms of root systems (14). On the other case either G = Gi or one of the three paths has length 1. If one path has length 1 hand, root systems can be generated starting from the Smith graphs (14, 25) and 10 D. CVETKOVlC AND P. ROWLINSON LARGEST EIGENVALUE OF A GRAPH 11

Note that among the graphs in Figure 3, the graph G9 has smallest index, approx- imately 2.007. Thus there is no graph with index in the interval (2,Ài(G9)).

ra A 2.2. Graphs whose largest eigenvalue does not exceed V2 + V5 FIGURE 2 In view of Theorem 2.1, the graphs of the title wil be determined once we have this makes it possible (25) to give elementary alternative proofs of some important found the connected graphs with index in the interval (2, V2 + VS); and we have theorems from (14). already noted that such graphs are trees. In order to describe the trees which arise, Let S be the set of all graphs (not necessarily connected) whose index does not let T( a, b, c) denote the graph with a vertex v of degree 3 such that T( a, b, c) - v = exceed 2. Cvetkovic and Gutman (29) determined explicitly the spectra of all graphs Pa U Pb U Pc. For a ;: 2, b;: 1, c;: 2 let Q( a, b, c) be the tree obtained from the path in S. One of their observations was that any eigenvalue is of the form 2cos(pjq)7r with vertices 1,2,.. .,a + b + c - 1 (in order) by adding pendant edges at vertices a for some integers p,q (q =f 0). It is interesting that the same conclusion follows from and a + b. If A and B are rooted graphs, Pn(A,B) denotes the graph obtained by an early result of L. Kronecker (56), as indicated in a review of (29) by J. H. Smith joining an endvertex of Pn to the root of A and the other endvertex of Pn to the root of B. All of these graphs are ilustrated in Figure 4. (cf. MR57, #5079). Kronecker's Theorem reads: let À be a non-zero real number which is a root of a monic polynomial p with integer coeffcients. If all roots of p The following classification theorem combines the results of several authors, and are real and in (-2,2), then À = 2cos27rr for some rational number r. includes implicitly the fact that no graph has index equal to V2 + VS. The index of a graph in S is either equal to 2 or is of the form 2 cos 7r j q for a positive integer q. THEOREM 2.4 (l1, 26) The connected graphs with index in the interval (2, V 2 + VS) Co spectral graphs from the set S have been studied in (29), too. For example, are precisely the trees of the following types. the graph Wn from Figure 1 is co spectral with the disjoint union of C4 and Pn+1' (a) T(a,b,c)for Note that for n = 0 we obtain the smallest pair of non-isomorphic cospectral graphs, consisting of Ki,4 and C4 U Pi. a = 1, b = 2, c)- 5 or The problem of deciding whether there exists a graph with given spectrum seems a = 1, to be intractable in the general case. Even more diffcult is the problem of construct- b)-'2, c)- 3 or ing all graphs with a given spectrum. However, if the eigenvalues (given) belong to a = 2, b = 2, c)- 2 or the segment (-2,2) both problems are easily solved and an appropriate algorithm is a =2, formulated in (29). b = 3, c = 3. The difference Ài - Àn between the largest and smallest eigenvalues is called the (b) Q(a,b,c)for spectral spread of a graph. In graphs from the set S the spectral spread is bounded above by 4. Since Àn = -Ài in bipartite graphs, bipartite graphs outside S certainly (a,b,c) E H2, 1,3),(3,4,3),(3,5,4),(4, 7, 4),(4,8, 5)J or have spectral spread greater than 4. Howeyer, there are only finitely many non- bipartite graphs outside S having spectral spread bounded by 4, as the following a)- 1, b;: b*(a,c), c)- 1 where (a,c) =f (2,2) and result of Petrovic shows. for a)- 3, THEOREM 2.2 (71) A connected graph has spectral spread bounded above by 4 if and b*(a,c) = 2 + c for a = 3, only if it is an induced subgraph of one of the Smith graphs or of one of the graphs in -1+ c for a =2. Figure 2. r+' / This theorem may be regarded as a generalization of Theorem 2.1. For the next result, due to Cvetkovic, Doob and Gutman, recall that a graph is A slightly weaker form of Theorem 2.4, without specification of the function minimal with respect to a property P if it has property P but none of its vertex- b*(a,c), was proved by Cvetkovic, Doob and Gutman (26). The function b*(a,c) was determined by Brouwer and Neumaier (11). The proof is similar in spirit to deleted subgraphs has property P. that of Theorem 2.1, but first it is necessary to verify that limn_oo.Ài(T(l,n,n)) = THEOREM 2.3 (26) There are exactly 18 graphs which are minimal with respect to limn_ooÀi(T(2,2,n)) = V2+VS; and to determine the conditions under which the property of having index greater than 2. These graphs, together with their indices /Ài(Pn(A,B)) is greater or less than Ài(Pn+i(A,B)). The relevant results on the sub- are shown in Figure 3. division of an edge are given in Theorem 4.5. ;":î~~ '~:_~sk-:)~~~:-*~~ß~~;~¿$j€f~~i~!;~#~~-'~:!~~:-1í:~!:~!!!!~~!!~li~#i'~¡m5fil2$~J~~~;:;æ~~1~r_3l=:;;:;;~;:;,::'t,~-\~::~:_::§ê'gE'~:E~:~1~;:!:~~!~::tli~Jt¡:._. -~- ~ - f . .

- ~ N Gi

tl ..() ~ o :: (), ~, o ~

Gs G6 a G7 ~ :; ~ viz zo

2.007

~o-GB Gg -0 FIGURE 3. The eighteen minimal graphs with index greater than 2.

~

'.

~ :; Q Gii GI2 GI3 tT "10 GI4 vi -- t: Q tT ~ 5 tI o 'I ;i Q :; ~ 'i ::

Gis GI6 GI7 Gis

FIGURE 3. (Continued) /: ~.. 14 D. CVETKOviC AND P. ROWLINSON LARGEST EIGENVALUE OF A GRAPH 15

2.3. limit points of graph indices 9 We say that the real number À is a limit point of graph indices if there is an infinite sequence of graphs Gn such that À = limn__coÀi(Gn) and the Ài(Gn) are distinct. The study of limit points of graph eigenvalues was initiated by Hoffman in the paper (50), where he determined all limit points less than ';2 + 0. It is clear from Section 2.2 that no number less than 2 is a limit point of graph indices; and we have noted that ';2 + vi is itself a limit point. Hoffman's result is as follows. THEOREM 2.5 (50) For n E N, let ßn be the unique positive solution of the equation xn+l = 1 + x + x2 + ... + xn-i, and let an = ß~/2 + ß;;il2. The numbers an (n E N) are the numbers less than ';2 + vi which are limit points of graph indices. Moreover 2 = ai -: a2 -: .,. and limn_co an = ';2 + VI. Each an is realized as the limit point of indices of trees which consist of a path Û with a pendant edge attached. In fact, the numbers an (n E N) are all the limit .t points in (2,';2 + vi) of spectral radii of symmetric matrices whose entries are .! () non-negative integers. In (50), Hoffman suggested that possibly there exists a real number À such that every number at least À is a limit point of graph indices. This turned out to be true with À = ';2 + VI, the value which Theorem 2.5 shows to be the smallest possible candidate. Indeed, Shearer (85) proved by direct construction that if ¡i :2 ';2 + vi then JL is a limit point of indices of trees. Limit points of eigenvalues other than the largest are studied in (36).

3. ORDERING OF GRAPHS J '" The ordering of graphs by index was proposed by Collatz and Sinogowitz (18) following their investigation of indices of trees. Lovász and Pelikán (62) suggested that among trees with a prescribed number of vertices, the higher the index the i more 'dense' the tree. In support of this view they proved that among trees with N n vertices the star Ki,n-i has largest index (vn -1) and the path Pn has smallest 0=N index (2cos7rj(n + 1)). In fact the first assertion is immediate from the observation .5 '0 (cf. Wang (97)) that the spectrum of a tree is symmetric about zero and satisfies the ~ :: relation Ài + ... + À~ = 2(n -1). We can go on to show that Ki,n-i is the only n- U '" vertex tree with index vn -1, for then the spectrum is (.J,o,O,.. .,O,-.J); :a 1l the adjacency matrix therefore has rank two, and so with appropriate ordering of 0. J ..(0 columns has the form (i, g) where each entry of B is /1. The tree is therefore a co complete bipartite graph and hence a star. ~ To show that Pn alone has smallest index, Lovász and Pelikán exploit the follow- -. ing expression (28, Theorem 2.12) for the characteristic polynomial of a graph G ¡i ~ with a bridge uv: ò;: "t '11 "U ~ ti qiG(x) = qiG-uv(X) - qiG-u-v(X). (3.1) If G is a tree other than a path then we can construct a tree G' with index smaller than G as follows. Choose vertices u, w in G such that deg( u) ? 2, deg( w) = 1 16 D. CVETKOviC AND P. ROWLINSON LARGEST EIGENVALUE OF A GRAPH 17

and d(u, w) is minimaL. Then u is adjacent to a vertex v not on the u - w path of Cs and a vertex of Cd-s then the index of the resulting graph decreases as s in G and we obtain G' from G by replacing the edge vu with vw. Since the increases (3 ~ s ~ (d 12)). graphs G - uv,G' - vw are isomorphic we have ÍJG(x) - ÍJG'(x) = ÍJG'-v-w(x)- So far we have been concerned primarily with trees and unicyclic graphs. The impetus for investigating the Ài-ordering of other classes of graphs came from two ÍJG-u-v(x) from (3.1). Now G - u - v is isomorphic to a spanning subgraph of G' - v - w, while repeated application of (3.1) shows that for any spanning subgraph quarters: Brualdi and Hoffman (7, p. 438) posed the problem of finding the maxmal H' of a tree H we have ÍJH(X) -( ÍJH'(X) for all x:; Ài(H). It follows that ÍJG(x)-( spectral radius of a (O,I)-matrix with a prescribed number of ones; and Cvetkovic ÍJG'(x) for all x:; Ài(G' - v - w), in particular for x = Ài(G'); hence Ài(G')-( (81, p. 211) asked how the index of a graph consisting of a fixed cycle and a chord Ài(G). Accordingly we have the following result. varies with the position of the chord. This second question was answered inde- pendently by Simic and Kocic (89) and (for a cycle of even length) by Rowlinson THEOREM 3.1 (18, 62, 97) Among the trees with n vertices (n:; 1), the star Ki,n-i (75), using entirely different methods. Simic and Kocic consider the more general alone has largest index and the path Pn alone has smallest index. class of n-vertex graphs consisting of k disjoint paths (k :; 2) between two vertices Similar arguments concerning characteristic polynomials were used by Simic in u and v. If the components of the principal eigenvector corresponding to vertices an analogous investigation of unicyclic graphs. The arguments make use of some in the i-th path are x~,xL...,X~I¡' then we have mi+mi+...+mk=n+k-2, general theorems on the change in index of a graph resulting from various modifi- xij = x5 =... = x~, xj = X~i¡_j (j = O,...,mi) and cations described in Section 4. xi.J+ i - Àixi,J+ i + Jxi. = 0 (j=0,...,mi-2). (3.2) THEOREM 3.2 (13, 86) Let Ktn-i denote the graph obtained from Ki.n-i by adding an edge. Among the uiiicyclic gráphs with n vertices, the graph Ktn-i alone has largest The recurrence relations (3.2) may be solved to give x~ as a function of mi,...,mk index and the cycle Cn alone has smallest index. ' and Ài. From the relation Ax = Àix we know that Àix~ = xi +... + xt, and this equation defines Ài as an implicit function of mi,..., mk. It is now a matter of cal- The second statement here is immediate from the fact that a unicyclic graph culus to show that if (say) m3,...,mk are held fixed then Ài is a strictly increasing has mean degree 2 and this lower bound for the index is attained precisely when function of Imi - mil. (Entirely analogous results hold for the class of graphs ob- the graph is regular. Further results concerning the Ài-ordering of unicyclic graphs tained by amalgamating the vertices u and v.) Setting k = 3 we can now answer are derived by Cvetkovic and Rowlinson in (32). For example, let Gm,n,r,s(n ~ m ~ Cvetkovic's question as follows: 1, m + n ~ 3, r ~ 1, s ~ 1) denote the graph obtained from Cm+n by attaching paths Pr+i,Ps+i at vertices distance m apart in Cm+n. (Attachment of a path is THEOREM 3.4 (89) Let Gn,k denote the graph consisting of an n-cycle 123... nl to- taken to mean attachment by an end-vertex.) Then Ài(Gm+i,n-i,r,s) -( Ài(Gm,n,r,s) gether with the chord lk (3 ~ k ~ n - 1). Then Ài(GIi,3):; Ài(GIi,4):; ... :; Ài(GIi,s) for 1 ~ m ~ n - 2. Let Eel (e ~ 3, f ~ 1) denote the graph obtained from Ce by where s = 1 + (nI2). attaching a path PI+l by an end-vertex. Then Ài(En+i,d) -( Ài(Gi,n.i,d-i). The fol- Thus the bicyclic Hamiltonian graphs with n vertices are distinguished by their lowing results of Li and Feng (59) may be applied to show that if r + s = d and 1 ~ indices; in particular GIi,3 alone has largest index and Gii,s alone has smallest index. r~(dI2)-1 then Ài(Gi,n,r,s)-(Ài(Gi,n,r+i,s-i); and if r-l~r-s~m:;1 then Simic (87) subsequently extended Theorem 3.4 by showing that the same conclusions Ài(Gm,n,r,s):; Ài(Gm,li,r+i,S-i). Moreover one can compare the indices of graphs ob- " hold if instead of adding to the cycle the chord lk we add an arbitrary connected tained by attaching two paths at the same vertex of a fixed cycle (cf. equation (4.2)). !-;." graph G with similar vertices ll, v identified with l,k respectively. (Vertices are sim- ~' i ilar if they lie in the same orbit of the automorphism group of G.) I.! THEOREM 3.3 (59) Let u, v be vertices of G such that d(u, v) = m. Let Gr,s denote the graph obtained from G by attaching a path of length r at u and a path of length s Rowlinson's approach to Theorem 3.4 was to introduce an algorithm which en- " " Ldi,ic;" at v. Then Ài(Gr,S):; Ài(Gr+i,S-i) under any of the following conditions ables the characteristic polynomial of a multigraph G to be computed recursively in terms of characteristic polynomials of local modifications of G. Suppose that G (i) m = 0, deg(u) ~ 1, and r ~ s ~ 1; has at least three vertices, let ll, v be distinct vertices of G and let m be the number , ¡' (ii) m = 1, deg(u) ~ 2, deg(v) ~ 2 and r ~ s ~ 1; of edges between II and v. Let G - (uv) denote the murÙgraph obtained from G by ~ it (iii) m:; 1, deg(u) ~ 2, deg(v) ~ 2, r - s ~ m and s ~ 1. deleting all edges between u and v, and let G* be the multigraph obtained from .~ G - (uv) by amalgamating u and v. Appropriate determinantal expansions yield the i~."'~ij,.'...,... These results too are proved by comparing characteristic polynomials (and using relation ' r¡ Ii equation (4.6)). In the situation (i) of Theorem 3.3, we may regard Gr,s as obtained

I.,".. from G by amalgamating u with an intermediate vertex of a path of fixed length ÍJG(X) = ÍJG-iuv)(x) + mÍJG-(x) + m(x - m)ÍJG-u-v(x) - mÍJG-u(x) - mÍJG-v(x), W~, , 'h' d = r + s: for 1 ~ r ~ (dI2), Ài(Gd-s,S) increases with s. Using different methods di (3.3) ~"."..' (d. Theorem 4.6), Simic (87) proved that if instead we amalgamate u with a vertex ~tt .,'",m :~; ij'. ,.l'" 18 D. CVETKOviC AND P. ROWLINSON LARGEST EIGENVALUE OF A GRAPH 19

Equation (3.3) is called the deletion-contraction algorithm. Note that if G is a graph then G* wil have multiple edges precisely when u and v have a common neighbour number of edges are prescribed.) Let See) denote the class of all graphs having in G; hence the multigraph setting. If we apply (3.3) to the graph Gii,k of Theo- preciselye edges, and let fee) denote the maxmal index of a graph in See). Brualdi rem 3.4 with u = 1 and v = k, we can eventually express the characteristic polyno- and Hoffman showed that when e = (~) and d ? 1, a graph G in S( e) has index mial of Gii,k in terms of characteristic polynomials of paths and cycles. (Here we fee) if and only if Kd is the only non-trivial component of G. They conjectured that when e = (~) + t, 0 -( t -( d, a graph G in See) has index fee) if and only if make repeated use of (3.1) and (3.3).) Now Pr,Cr have characteristic polynomials Ur(lx),2Tr(lX) - 2 respectively, where T"Ur are Chebyshev polynomials of the first the only non-trivial component of G is the graph Ge obtained from Kd by adding and second kind (28, p. 73). Accordingly the characteristic polynomial of Gii,k has one new vertex of degree t. By applying perturbation-theoretic methods to adja- cency matrices, Friedland (43) proved that the conjecture is true for t = d - 1 and ~ an expression in terms of Chebyshev polynomials; for even n this expression can :111 further that there exists K(t) ? 0 such that the conjecture is true for d ~ K(t). Sub- ,Ii' be analyzed to yield the conclusion of Theorem 3.4. Similar techniques were used sequently, Stanley (92) proved that f (e) :: l( - 1 + vI + 8e), with equality precisely .;¡;l~ ; by Bell and Rowlinson (3) in an investigation of tricyclic Hamiltonian graphs (cycles ~ with two chords). They showed first that if such a graph has n vertices and max- when e = (~). Friedland (44) refined Stanley's inequality and thereby proved that the ~~.~ mal index then the two chords have a vertex in common. Let lL(h,t,k) denote the conjecture holds when t is 1, d - 3 or d - 2. The conjecture was finally proved true in ;~ general by Rawlinson (76). Since the components of the principal eigenvector of Ge 1 index of the graph Gh,t,k (h ~ 1, t ~ 0, k ~ 1, h + t + k + 3 = n) consisting of an n- ~ take only three values, Ài(Ge) is a root of a cubic equation, and one can show easily ~ cycle 123... n1 together with chords joining vertex 1 to vertices h + 2 and n - k. i The results on Ài-ordering are (i) if 1:: k :: t then lL(h,t,k) -( lL(h,k - 1,t + 1), that fee) = d -1 + € where 0 -( € -( 1 and é + (2d -1)('2 + (d2 - d - t)€ - t2 = O. '.R._.I' ~ The starting point for investigations of See) is the observation that the maxmal ~ (ii) if k ~ t ~ 1 then lL(h,t,k) -( lL(h,t - I,k + 1), (iii) if 2:: h:: k then lL(h,O,k)-( ~' index of a graph in S( e) is attained by a graph having a stepwise adjacency matrix, ¡.(h - 1,0,k + 1). It is now easy to identify the unique graph with maxmal index. that is an adjacency matrix (aij) which satisfies the condition ~~~. i ~ i THEOREM 3.5 (3) Among the tricyclic Hamiltonian graphs with n vertices (n ~ 5), tL the graph Gi,O,Ii-4 alone has largest index. 'ii (*) if i -( j and aij = 1 then ahk = 1 whenever h -( k :: j and h:: i. jU It should not prove too diffcult to determine the tricyclic Hamiltonian graphs J with smallest index. Another open question concerning indices of Hamiltonian I~ To see this, suppose that A is the adjacency matrix (aij) of a graph in See) and ~r! graphs relates to the family 1111 of maxmal outerplanar graphs on n vertices. Evi- ~L \1 Ax = Ài(A)x where IlxlI = 1, x = (xi,...,xii) and vertices are ordered so that Xi ~ .. dence from "Graph" suggests that the fan Ki 'iPIi-i is the unique graph in 1111 with X2 ~... ~ Xii ~ O. If for example apq = 0 and ap,q+i = 1 where p -( q then take I~ largest index and that P; is the unique graph in 1111 with smallest index. (The graph A' to be the matrix obtained from A by interchanging the (p,q) and (p,q + 1) ~ Ki 'i Pii-i is obtained from Pii-i by adding a vertex adjacent to each vertex of Pii-i; ~ entries and interchanging the (q,p) and (q + 1,p) entries. Then Ài(A') - Ài(A) ~ h~ and the graph P; is obtained from Pii by joining vertices which are distance 2 apart xT(A' - A)x = 2xp(xq - xq+i) ~ O. Similar arguments deal with the case in which ~~ in Pii') In support of this conjecture, Rawlinson has proved the following. N ~ apq = 0, ap+i,q = 1 and p + 1 -( q; and repeating the procedure as necessary we I THEOREM 3.6 (80) Let gii (n ~ 4) denote the class of maximal outerplanar graphs can obtain a stepwise matrix B such that Ài(B) ~ Ài(A). Refining the arguments we ff.* I which have n vertices and no internal triangles. Then Ki 'i Pii-i is the unique graph can show that Ài(B)? Ài(A), and it follows that every graph in See) with maxmal ~" ~ in gii with maximal index and pi; is the unique graph in 911 with minimal index. index has a stepwise adjacency matrix. In the special case that e = (Ü, Brualdi and ~R Hoffman are able to construct from B in four stages a matrix E such that Ài(B) :: J!I Here an internal triangle of the maximal outerplanar graph G is a 3-cycle which !.;i.:iii¡ Ài(E):: d - 1. Thus Kd has maxmal index in See); moreover it can be shown that iil has no edges in common with the unique Hamiltonian cycle 2 of G. If G E gii when Ài(B) = d - 1, B has the required form, namely (~-I ~), where each entry of ~. then the graph G - E(2) consists of a tree G** and two isolated vertices; and jf ~ ~ ~ G = Ki 'i Pii-i then G** = Ki,Ii-3 while if G = P; then G** = PIi-2. Now in view J is 1. (A short proof of this result is given below.) The general case e = (~) + t öf Theorem 3.1 it is natural to ask whether Ài(Gi) -( Ài(G2) whenever Gi,G2 E gii (0 -( t -( d) proved less tractable (see (42)): here a graph with maximal index has an adjacency matrix of the form and Ài(Gi*) -( Ài(Gi*). A pair of 10-vertex graphs exhibited in (80) shows that this / is not always the case; in other words the Ài-orderings of the graphs G E g10 is in- consistent with the Ài-ordering of the trees G**. The proof of Theorem 3.6 extends ¡J - I C OJ the techniques used by Brualdi and Hoffman in a paper (12) which provides partial A' = cT 0 0 answers to the questions they posed nearly ten years earlier (7, p. 438). As far as o 0 0 graphs are concerned, the basic problem here is to determine those graphs which have maximal index when just the number of edges in prescribed. (Note that The- orems 3.1 to 3.6 pertain to graphs for which both the number of vertices and the where J - I has size d x d and c T = (1,..., 1,0, . ..,0) with t non-zero entries. To prove this, Rowlinson (76) first shows that the condition (*) may be strengthened by 20 D. CVETKOviC AND P. ROWLINSON LARGEST EIGENVALUE OF A GRAPH 21

adding the requirement

( ** ) if (i) h -( p -( q -( k, d; and (ii) ahk = 1, alij = 0 whenever j ? k; aik = 0 whenever i ? h, ~r~~' p edges Mi ,'I ¡l~' and (ii) apq = 0; apj = i whenever p -( j -( q; aiq = i whenever i -( p, ,fi, then p + q :: h + k + L -a,~¡Ii '. ~\;~¡ ":i\! \r~~. i!!1 For a matrix A other than Æ satisfying (*) and (**) he compares indices by consid- ering the relation PC,., p. ql ~li ()'i(A') - Ài(A))xTxl = xT(AI - A)x' (3.4) l,ij "i,i where A'xl = Ài(A')x', x':: 0 and IIx'll = 1. Since xTx'? 0 the sign of Ài(A') Wi J¡~ :t! -Ài(A) is determined by the sign of the biquadratic form on the right hand side of :,",,~( (3.4). This is expressible as a - ß where each of a and ß is a sum of r terms of the ¡~¡ form xixj + XiXj, and 4r is the number of non-zero entries in AI - A. The proof of the Brualdi-Hoffman conjecture requires a delicate analysis of these terms which l exploits the condition (**). ~ We note in passing that equation (3.4) enables us to deal immediately with the special case in which e = (~). If A' = (~-I ~) and A is any other stepwise adjacency 4 :¡ matrix of the same size then a is the sum of r terms Xi xj + xi X j for which i -( j :: d ~1~"i,,',L~l . Bl,., p. q) 'if: and ß is the sum of r terms xixj + XiXj for which i -( j and j :: d + 1. Since Xl =

'~;\ ... = xd, xi = 0 for i? d, and Xi :: X2 :: ... :: Xd :: Xd+i ? 0 we have FIGURE 5. i I (Ài(A') - Ài(A))xTxl = a - ß:: rXi(2Xd - Xd+i)? O. It follows that, to within isolated vertices, Kd is the unique graph with (~) edges (c) e = (~), d:: 4, and G = Kd, and maxmal index. The general result is the following. (d) e = (~) + 1, d :: 5, and G = Hd, THEOREM 3.7 (12, 76) Let e = (~) + t where d? 1 and 0:: t.. d. For t ? 0 let Ge (e) e = (~) + t, 1.. t.. d, d:: 3 and G = Ge. be the graph obtained from Kd by adding one new vertex of degree t. If G is a graph We now return to the situation in which both the number of edges and the num- with maximal index among the graphs with e edges then G has a unique non-trivial ber of vertices are prescribed. Let 'H( n, e) denote the class of all connected graphs component H; H = Kd when t = 0 and H = Ge when t? O. with n vertices and e edges. In seeking the graphs with maxmal and minimal index in 'H( n, e), we may suppose, in view of Theorems 3.1 and 3.2, that e = n + k where Note that Kd is Hamiltonian when d :: 3 and Ge is Hamiltonian when d:: 3 and k:: 1. ", 1.. t.. d. Accordingly to find the Hamiltonian graphs in See) with maxmal index I ~,~',', ",'1,'' In complete generality, this problem is harder than the corresponding problem ¡'i" it suffces to consider the case e = (~) + 1. For d? 1 let Kd denote the graph ob- for See). As far as minimal index is concerned we have just the following result ~i,,!. tained from Kd by deleting an edge; and for d :: 4 let Hd denote the graph obtained ';"~"., ",,' of Simic, proved by extending to 'H ( n, n + 1) the techniques he used in proving :. :¡j from Kd by adding one new vertex adjacent to precisely two vertices of degree d - 1 Theorem 3.4. " in Ki. Then Hd has (~) + 1 edges and is Hamiltonian for d? 4. The techniques of !J THEOREM 3.9 (88) Among the bicyclic graphs with n vertices, n:: 7, there are pre- 1m (76) may be extended to show that (for d:: 4) Hd has the second largest index ~, '~."" 1r: of any connected graph with (~) + 1 edges, and that Hd is unique in this respect. cisely two graphs with minimal index. In the notation ot,Figure 5, one is the graph P(k,n + 1 - 2k,k) and the otheris the graph B(k,n + 1 - 2k,k), where k = rnj3l- ~ Indeed, Rowlinson (77) shows that for d ? 4 the only graphs with e = (~) + 1 and ~¡ index greater than Ài(Hd) are, to within isolated vertices, the graphs Kd U K2 and 'rm If À(m,p,q) denotes the index of P(m,p,q) then (as we noted in the preamble to Ge. Small values of e are treated separately and the complete picture is as follows. Theorem 3.4) for fixed m,À(m,p,q) is a strictly increasing function of Ip - ql (87). THEOREM 3.8 (77) Let G be a Hamiltonian graph with e edges, e :: 3. If the index If p(m,p,q) denotes the index of B(m,p,q) then for m .. qand fixed p,p(m,p,q) .. ,. ~,I.:ii of G is maxmal then one of the following holds: is a strictly increasing function of q (88). ~~ '; The remaining results in this section are concerned with graphs in 'H(n,n + k) ~ (a) e = 4 and G = C4, '~~i¡ , ': (b) e = 7 and G is the unique maxmal outer planar graph on 5 vertices, which have maxmal index. Brualdi and Solheid (13) show that again such a graph :it t~ ~,:,J

'~

,L. 22 D. CVETKOviC AND P. ROWLINSON LARGEST EIGENVALUE OF A GRAPH 23

G has a stepwise adjacency matrix (aij)' Thus aiz = an'" = ain = 1 and G has a (i) G = Gii,k if n ,g(d), spanning star with central vertex 1. Let G* denote the graph induced on vertices (ii) G = Hii,k or Gn,k if n = g(d), 2, .. ., n by the remaining k + 1 edges of G. In all known cases, a graph G with (iii) G = Hii,k if n? g(d), maxmal index is one of two types. The first type, denoted by Hn,k (0:: k :: n - 4), is the graph G for which G* consists of a star and isolated vertices. To describe where g(d) is defined by equation (3.5). the second tye, denoted by Gn,ki let k + 1 = (di-i) + t (0:: t :: d - 2): if t = 0 then It follows that for e = n + k = n -1 + (di-i), Gii,k is the unique graph with max- G* consists of Kd-i and isolated vertices, while if t ? 0 then the only non-trivial mal index in 'H(n,e) whenever n, 60 or whenever e ~ 2n - 47; moreover Theorem component of G* is the graph obtained from Kd-i by adding one new vertex of 3.12 provides an improvement on Yuan's upper bound (Theorem 1.6) for the index degree t. The graphs Hn,k, Gn,k may also be described by assigning k + 1 ones to the of a graph in 'H(n,e), e = n -1 + (di-i). Bell's methods again make use of equation triangle of positions (i,j), 1, i , j, in a stepwise adjacency matrix. In the first case, (3.4) and represent a further refinement of the arguments in (76), but in this special they are assigned to the first available row; in the second case they are assigned case an analogue of condition (**) is not needed. column by column. Note that Hn,i = Gn,i. We shall see later that for certain other values of nand k we have ),i(HIl,k) = ),i(Gn,Ù For a specific value of k there are only finitely many possibilities for G*, and this 4. GRAPH OPERATIONS AND MODIFICATIONS fact enabled Brualdi and Solheid to undertake an exhaustive analysis of the cases in We begin by considering the indices of graphs constructed in various ways from which k :: 5. Then the characteristic polynomial of G has the form xn-r fr(x) where two graphs Hand K. Let IV(H)I = s, IV(K)I = t. First, the disjoint union HÜK deg(fr) = r :: 6 and in fr the coefficients are linear functions of n. For sufficiently clearly has index equal to maxpi(H),),i(K)). It is also straightforward to deal large n, the graphs in 'H( n, n + k) with maximal index can then be identified. When with the sum H + K, product H x K and strong product H *K, each of which has k :: 2, these graphs are known for all n and we record the result as follows. vertex set V(H) x V(K). Vertices (Ub vi),(uz, vz) are adjacent in H + K if and only THEOREM 3.10 (13) Among the bicyclic graphs with n vertices the graph Gn,l alone if either Ui = Uz and Vi rv Vz or Ui rv Uz and Vi = vz; adjacent in H x K if and only has maxmal index; and among the tricyclic graphs with n vertices, the graph Gn,z if Ui '" Vi and Uz rv vz; and adjacent in H *K if and only if they are adjacent in H + alone has maxmal index. K or H x K. If H,K have adjacency matrices A,B respectively then H + K,H x

Brualdi and Solheid prove also that for k E P,4,5) there exists N(k) such that K,H*K have adjacency matrices (A (9li) + (Is (9 B), A (9 B, (A (9 Ii) + (Is (9 B) + for n? N(k), Hn,k is the unique graph in 'H(n,n + k) with maxmal index; and they (A (9B) respectively, and it follows that ),i(H + K) = ),i(H) + ),i(K), ),i(H x K) conjecture that the same conclusion holds for all k ~ 3. This conjecture was con- = ),i(H)),i(K), ),i(H*K) = ),i(H)),i(K) + ),i(H) + ),i(K). For a general setting for firmed by Cvetkovic and Rowlinson (33), and the essential ingredient for their proof these results, see (28, section 2.5J. is the biquadratic form given in equation (3.4). Now let u be a vertex of H, va vertex of K. If HuvK denotes the graph obtained from HÜK by adding the edge uv then THEOREM 3.11 (33) Let 'H(n,n + k) denote the class of all connected graphs with n l~ vertices and n + k edges. For k ? 2 there exists N(k) such that for n ? N(k), Hn,k is aiHuvK(X) = ØH(X)(/J(x) - aiH-u(X)aiK-v(X). (4.1) the unique graph in 'H(n,n + k) with maxmal index. i~ If G is obtained from HÜK by amalgamating u and v then ~~ ,,, Some isolated results on the ),i -ordering of 'H( n, n + k) (k :: 5) are given in (13)

I.~~..' and (33) with a view to estimating N (k). Bell (2) has pursued this question in the aiG(x) = aiH(X)ØK-v(X) + ØK(X)ØH-u(X) - XaiH-u(X)aiK-v(X), (4.2) ili case that k + 1 = (di-i) (5:: d :: n -1). (This case corresponds to the special case The relations (4.1) and (4.2) may be established by using appropriate determInantal m e = (~) for See) considered by Brualdi and Hoffman (12).) Bell not only gives N(k) ~.~ expansions: see (28, Theorem 2.12) and (75, Remark 1.6). In either case, the spec- ~I as an explicit function g(d) but also shows that when n, g(d) the graph Gn,k is ~ ,~ trum (and hence the index) of the graph concerned is determined by the spectra .' the unique graph with maximal index in 'H(n,n + k). Moreover if g(d) is an integer of H,K,H - u and K - v. Accordingly it is of interest.to investigate further the then ),i(GIi,k) = ),i(lfii,k) when n = g(d). In fact, Ii characteristic polynomial of a graph which is modified by the removal of a vertex. ~; m 1 32 16 Let G be a graph with vertices 1,2,..., n and adjacency matrix A. Let A have ~l.~ g(d) = Zd(d + 5) + 7 + d _ 4 + (d _ 4)Z (d ~ 5) (3.5) ai! spectral decomposition ¡.iPi + ... + ¡.niPni, and let ei,.. .,en comprise the standard orthonormal basis of R" . and so n = g(d) if and only if (n,n + k) E t(60,69), 88), 85)). The complete I~ (68, (80, Godsil and McKay (46) pointed out that an expression for øG-u(x), u E V(G), result is the following. l~: may be obtained by expressing in two ways the (u,u)-entry of the matrix generat- THEOREM 3.12 (2) Let k + 1 = (di-i), where 5:: d :: n -1, and let G be a graph ~ ing function Lr:o x-k Ak. On one hand, Lr:o x-k Ak = (I - x-i A)-i with (u, u)- ~ with maximal index in 'H(n,n + k). Then entry xaiG-u(x)/lG(x), because aiG-u(x) is the (u,u)-cofactor of xI - A. On the 24 D. CVETKOviC AND P. ROWLINSON LARGEST EIGENVALUE OF A GRAPH 25

other hand, Ak = 'L':i p.f Pi and Pi has (u, u)-entry afu where aiu = IP¡eu I; hence of P.i then sum(Pi) = nß¡, k(t) = 'L7~l'L'lonßfp.ftk = 'L':i nßfI(l- tP.i) and the (u, u)-entry of 'L~o x-k Ak is 'L':i 'L~o x-k p.f afu, which is expressible as ~ 'L':i afu/(I- X-iP.i). Therefore, f m nß¡ ì. ~ CPG( X) = (-It cpo( - x-I) ì. 1 - n f; x + 1 + p¡ f . ( 4.5) ~¡ ,1i, F~ m 2 11t , ,II CPG-u(X) = CPG(X) L ~. ( 4.3) The numbers ßi,...,ßm are called the main angles of G. In view of equations (4.4) 1"-''...1''l" ~ ¡=i x- Pi and (4.5) we have the following result. W('~i It:1 Note that cos-i(aiu) is the angle between eu and the i-th eigenspace of A. The ~. THEOREM 4.2 The spectrum of H'V K is determined by the spectra and main angles ß numbers aiu,...,amu are commonly called the angles of G at u, abusing terminol- ofHandK. ogy. In view of (4.1), (4.2) and (4.3) we conclude the following. j~ For the remainder of this section we discuss various modifications of a graph G, ~" THEOREM 4.1 Let u be a vertex of the graph H, v a vertex of the graph K. If the using the above notation for angles and orthogonal projections. We have already graph G is obtained from H U K either by amalgamating u and v or by adding the I;~i seen from equation (4.3) that for any vertex u of G, Ài(G - u) is determined as bridge uv then the spectrum of G is determined by the spectra of Hand K, the angles the largest root of CPG(x) 'L7~i afu/(x - Pi). If again we denote by Gu the graph .~.1' ..'..1,...'. of H at u and the angles of K at v. obtained from G by adding a pendant edge at u then as a special case of (4.1) we have 00 ;,:¡ ìl,¥' i' Let us write Hu for the graph obtained from H by adding a pendant edge at ver- iPGu(x) = xiPG(x) - CPG-u(x), (4.6) ~'i, tex u. We note in passing that Zhang, Zhang and Zhang (104) extend Theorem 3.3

I".". by showing that if CPHu(X) -( CPHv(X) for all x? Ài(Hv) then Ài(HuwK)? Ài(HvwK) a relation which is easy to prove directly. Accordingly ~., for all vertices w of K. iö': We now turn to another means of combining two graphs Hand K: the join (or CPG,(X) = iPG(X) fì. xm - f; 2X ~upiì. f (4.7) .1~.,...... I..~,;I complete product) H'V K is obtained from H U K by joining every vertex of H to

, ' every vertex of K. In other words, H'VK is the complement of HUK, and this . ',',i.), . i~:! makes it possible after a little work (28, Theorem 2.7) to express the characteristic and so Ài(Gu) is determined by the spectrum of G and the angles aiu,...,amu' ~~I' II polynomial of H'V K in the following way: More generally we have the following observation...... '.'~-1: THEOREM 4.3 For any vertex u of G, the spectra of both G - u and Gu are deter- CPW\lK(X) = (-iyCPH(X)~( -x -1) + (-iyCPK(X)CPii -x -1) l...,c.S mined by the spectrum of G and the angles of G at u. !.I.. ..-.:.,.,Ii...,.il..,.. '~~ + (-ly+t+icpii(-x -1)cpï((-x -1). (4.4) ;1; If, in the construction of Gu from G, the new pendant vertex is labelled 0 then : -ii; If H is regular of degree d and K is regular of degree e then it follows that CPG, (x) = det(xl - Ao - B) where Ao is the (n + 1) x (n + 1) matrix (0 A) and the m+ .'~; Ài(H'VK) = itd + e + vI(d - e)2 + 4st1 because then the spectrum of H consists only non-zero entries of B are ones in positions (0, u) and (u, 0). In the case that G ~~ ~~ 'Ii! of s - 1- d, -À2(H) - 1,..., -Às(H) - 1 and the spectrum of K consists of t -1- e, is connected and not a complete multipartite graph, Bell and Rowlinson (4) sought ~. -À2(K) - 1,..., -Àt(K) - 1. In general however we need to investigate further the an explicit expression for Ài(Gu) in terms of the spectrum and appropriate angles ~li.i characteristic polynomial of the complement of a graph (the result of a unary graph of G by expressing the largest root of det(xl - Ao - (B) ((? 0) as a power series t~ ,:'. i~ operation). in (. For this to be of any use we require convergence at ( = 1, and it turns out that ~¡F.i'. Il If as before G has adjacency matrix A then G has adjacency matrix J - I - A the radius of convergence of the series does exceed 1 for large enough Pi - P2. If ..- .... for example Pi - P2 ? 4 then Ài( Gu) = pi + 'L~i Ck where the Ck are recursively i~: and characteristic polynomial det((x + 1)1 - J + A), where each entry of J is 1. defined functions of the 2m invariants Pi, a¡u of G (l;= 1,...,m). In fact the Ck I~. Thus if(x) = det((x + 1)1 + A) - sumadj((x + 1)/ + A), where sumadj( ) denotes ~ : are just the so-called perturbation coefficients which arise in the analytical theory ¡~. the sum of all entries of the adjoint matrix. It follows (cf (34, Theorem 5)) that 11' of matrix perturbations (57, §§11.5, 11.6) applied to the linear perturbation Ao + WI .,6.,..; (B(( E C). Here, if xo denotes an eigenvector of Ao corresponding to pi, one finds ar(x) = (-I)ncpG(-x -1) - (-It(x + 1)~i k (~) CPG( -x -1) sufficient conditions for the existence of analytic functions pi(() = pi + 'L~i Ck(k, ¡" G X + 1 .I...~ x(() = Xo + 'L~i Xf(k such that (Ao + (B)x(() = pi(()x((). Rowlinson (78) applied where fG(t) = sumadj(1 - tA)/det(1 - tA) = sum .¡~n (I - tA)-i = sum'L~oAktk.Now this theory in the case of a graph G + uv obtained from a connected graph G by sum(Ak) = 'L7~ipfsum(Pi) and sum(Pi) = 'LuvPieu' Piev = IP¡jI2 where j denotes joining two non-adjacent vertices u and v. Not surprisingly, more invariants of G are J~ l!! the all-l vector. Thus if cos-1(ß¡) is the angle between j and the eigenspace required to determine Ài(G + uv) in this general case: in addition to the spectrum ~ LARGEST EIGENVALUE OF A GRAPH 27 26 D. CVETKOVlC AND P. ROWLINSON the same set of vertices. Simic (87, Theorem 2.4) deals in this way with a graph and t eh ang I es(. aiu,aiv - i1 - ,...,m) . we iii require (ml IUV"",luV h -i( were cosIiI). IUV is ht e G' obtained from G by splitting a vertex of G: if the edges containing v are vw angle between Pieu and Piey (defined when these vectors are non-zero). If pi - P2 (w E W) then G' is obtained from G - v by adding two new vertices Vi, V2 and is large enough (say, greater than 4) then Ài(G + uv) is the sum of a convergent edges ViWi (wi E Wi) V2W2 (W2 E W2) where WiUW2 is a non-trivial bipartition series pi + L~i Ck where the Ck are recursively defined functions of the invariants ofW. Pi,aiU,aiv,lt¿ (i = l,...,m). A convergent series may of course be used to compute THEOREM 4.5 (87) If G is a connected graph and G' is obtained from G by splitting the index to any degree of accuracy, but if we require merely an estimate for the a vertex then Ài(G') ~ Ài(G). index then for an upper bound we may turn to an algebraic theory of perturbations (99, Ch. 6). Maas drew attention to this theory in a paper (64) which treated both Finally we consider the case in which G' is the graph Gu,v obtained from G by G + uv (where G is connected) and the addition of a bridge between two disjoint subdividing the edge uv: thus G' is obtained from G - uv by adding a new vertex connected graphs. The idea is to consider à + ÉP where P is an appropriate projec- wand edges wu, wv. Note first that the subdivision of an edge does not necessarily tion, à = -A - (Ài(B) + b)I and É = (Ài(B) + b)I - B. Here b is chosen positive result in a change of index: if G is an n-cycle Cn then always Ài (Guv) = Ài (G) = to ensure that É is a positive matrix and hence that ÀIi(à + ÉP) :: Àn(à + É). Since 2, and if G is the (n + 5)-vertex graph Wn depicted in Figure 1 then Ài(Gu,v) = à + É = -A - B, we have Ài(A + B):: -ÀIi(à + ÉP). The parameter b is chosen Ài(G) = 2 for any non-pendant edge uv. path of G as a walk vovi,...,vk(k:: 1) to optimise this upper bound for the index of the perturbed graph. If A and A + B Hoffman and Smith (53) define an internal are the adjacency matrices of G and G + uv (where G is connected), and if P is such that the vertices vi,..., Vk are distinct, deg(vo) :/ 2, deg(vk) :/ 2 and deg(vi) = 2 an appropriately chosen projection onto the eigenspace of pi then we obtain the whenever 0 .( i .( k. Thus an internal path gives rise to either a subgraph Wk or a following. k-cycle with one pendant vertex attached. By constructing a suitable positive vector the connected graph G. Then y in the case that the edge uv lies on an internal path of G, Hoffman and Smith THEOREM 4.4 (64) Let u, v be non-adjacent vertices of prove the following. Ài(G + uv):: Ài(G) + 1 + b -, where b:/ 0 and = b(l + b)(2 + b) = G _ G THEOREM 4.6 (53) Let uv be an edge of the connected graph G. I (aiu+aiv)2+ó(2+Ó+2aiUaiy) pi() P2( ). (i) If uv does not belong to an internal path of G and if G f- Cn then Ài (Gu,v) :; Ài(G). Analogous results for the deletion and the relocation of an edge are obtained (ii) If uv belongs to an internal path of G and G f- Wn then Ài (Gu,v) ~ Ài (G). by Rowlinson in (79). Further, Maas (64) deals with HuvK as a perturbation of H U K and Rowlinson (79) deals with K i \7 G as a perturbation of G (an example of a global rather than a local modification of G). The results in these cases are 5. THE LARGEST EIGENVALUE OF RANDOM GRAPHS somewhat technical and we omit the details. A special case of the construction This short section has been included for the sake of completeness. There has H uv K is the addition of a pendant edge. Here we can simply use equation (4.7) been a remarkable development of the theory of random graphs in recent years: to show that if pi(Pi - P2):/ 1 then Ài(Gu).( pi + tu where tu = aiu/(Pi - (pi- anal- several topics from the theory of (usual) graphs have prompted the study of P2)-i) (79, Remark 5.3): since the eigenvalues of G interlace those of Gu, it suffices ogous questions in random graphs, and a review of eigenvalues of random graphs to check that rpGu (pi + tu) :/ O. may be found in Section 3.7 of the monograph (27). We mention here just two re- In our consideration of estimates for the index of a modified graph we have so sults concerning the largest eigenvalue of undirected random graphs. .:1 far discussed only upper bounds. Lower bounds are readily obtained from Rayleigh Let Gn,p denote a random graph on n vertices, each pair of vertices being con- quotients since for any real symmetric matrix A' we have Ài(A') = SUPtZT A'z/zTz : nected by an edge with probability p (0 ~ p .( I). Juhász proved the following result. z f OJ' In particular if x is the principal eigenvector of A then Ài(A + B) :: Ài(A) + probability 1. xTBx; for example, Ài(G + uv):: Ài(G) + 2aiuaiv' Accordingly the index of a mod- THEOREM 5.1 (55) Lim,l-oo(l/n)Ài(Gn,P) = p with / ified graph can be restricted to a certain interval, and the effects on the index of two The second result is an application of the method of bounded differences de- different modifications can be compared if the lower limit of one interval exceeds scribed by McDiarmid in (66). In order to apply Lemma 3.3 of that paper we need to the upper limit of the other. 'êl observe that if the graphs G and G' differ in only one edge then IÀi(G) - Ài(G')1 :: We can also find upper bounds for the index of a modified graph G', with adja- 1. We may assume that G is connected and G' is obtained from G by deleting the cency matrix A' as follows: if y is a positive vector and p a scalar such that A'y :: py edge ij. If G has adjacency matrix A and principal eigenvector x = (xi,X2,...,xn)T and A'y f py then Ài(A').( P (67, Theorem 1.3.1). This is often useful when G' is then Ài(G'):: xT Ax - 2XiXj = Ài - 2XiXj :: Ài - xr - xJ:: Ài -1. The hypotheses obtained from G by the introduction in some way of an additional vertex, because of (66, Lemma 3.3) are therefore satisfied and we deduce the following. the perturbation theories described above apply directly only when G and G' have 28 D. CVETKOviC AND P. ROWLINSON LARGEST EIGENVALUE OF A GRAPH 29

THEOREM 5.2 If E(Ài) denotes the expected value of Ài = Ài(GIl,P) then for t :; 0, may be used to estimate the rate of growth of some combinatorial sequences. This applies in particular to the transfer matrix method used in several enumerations in Pr(IÀi - E(Ài)1 ;: t) :S 2exp t -2t2 / (~) ) . physics and chemistry (cf. for example, (28), pp. 245-251). We mention in passing an application to tournaments (98) (see also (6) or (28, p. 226)): when ranking the participants of a tournament (a complete directed graph) 6. APPLICATIONS one can use coordinates of the principal eigenvector. For any graph G, let E(G) = Ài(G) - d(G), where d denotes the mean degree. In this section we give a brief commentary on some applications of the results de- Collatz and Sinogowitz (18) proposed E as a measure of irregularity: note that scribed in previous sections. We divide the applications into two groups: applications by Theorem 1.1, E;: 0 with equality if and only if G is regular. Bell (5) shows to other mathematical problems (mainly again in graph theory) and applications to that the largest possible value of E for an n-vertex connected graph lies between other disciplines (physics, chemistry, computer science, geography). ln - î + 21n and ln - 1 + Iln. A natural measure of irregularity is the variance of the vertex degrees, that is v(G) = (lln)'£?=i(di - d)2. Rowlinson (80) gave exam- 6.1. Applications within mathematics ples of maximal outerplanar graphs Gi, G2 such that v( Gi) = v( G2), E( Gi) :; E( G2); Theorems about graph spectra can sometimes be used to prove results in graph and of associated trees Gi*,Gi* such that v(Gi*) = v(Gi*), E(Gi*) -( E(Gi*). Sub- theory and combinatorics which themselves make no mention of eigenvalues. Such sequently, Bell (5) established that E and v are actually inconsistent as measures of means of proof are often referred to as 'spectral techniques'. Well-known instances irregularity in respect of the graphs Gll,k and HIl,k of Theorem 3.12. (Note that for are structure theorems for strongly regular graphs and existence theorems for Moore prescribed numbers of vertices and edges, E-ordering coincides with Ài-ordering.) graphs. Indeed spectral techniques often prove their worth in extremal graph theory: Finally we note that Pötschke (73, 74) discusses the rôle of Ài in the graph iso- see Chapter 7 of the monograph (28). Some further examples are given below. morphism problem. Upper and lower bounds on the index of a graph, as described in Section 1, may be combined to provide inequalities relating non-spectral invariants. For example 6.2. Applications in other disciplines Wilf (100) has combined Theorems 1.5 and 1.7 to obtain an upper bound for the The theory of graph spectra has several applications in physics and chemistry: see chromatic number of a graph G with n vertices and e edges: (28, Chapter 8) and (27, Chapter 5). For example, a membrane (with fixed bound- ary) may be represented by a lattice graph whose eigenvalues determine the har- ,( G) :S 1 + V 2e (1- ~). monic oscilations of the membrane: the largest eigenvalue corresponds to the os- cillation with least energy. In HückeI's theory of molecular orbitals, the eigenvalues If G is connected and we use Theorem 1.6 instead of Theorem 1.5 then we obtain of the graph G representing the carbon skeleton of a hydrocarbon molecule deter- mine the quantum energy levels for the molecule: again the largest eigenvalue of ,( G) :S 1 + v2e - n + 1. G corresponds to the lowest energy state. Several physical and chemical properties Theorem 1.10 has been used in (70) to prove and extend Turan's theorem (cf. of saturated hydrocarbons (e.g. viscosity, surface tension, boilng point, density etc.) depend on the "extent of branching" of G. A number of topological indices (i.e. (28), pp. 221-222). See also (40), where Theorem 1.2 has been used to obtain a lower bound on the number of triangles in a graph. graph-invariants) have been proposed and studied as a means of treating branching As a third application we mention the problem of determining the number of in a quantitative manner. Among them we find the largest eigenvalue of G, sug- walks in a graph. Let Nk be the number of walks of length k in a graph G with gested by Cvetkovic and Gutman in (30): there the asymptotic formula (6.1) is used adjacency matrix A. It follows from the spectral decomposition of Ak that to justify empirical findings that Ài(G) is a suitable measure of branching. Calcu- lation of the largest eigenvalue in a chemical context features in (72), and further il methods for calculating graph indices are described in (58). Nk = ¿ciÀr The idea of the index of a graph as a measure of btánching has also been ap- i=i plied in a quite different area, namely the theory of algorithms-in particular algo- where the Ci are quantities which depend on the eigenvectors of A but not on k (cf. rithms for so-called N P-problems. An N P-problem is intractable in the sense that (28), p. 44). We immediately derive the following asymptotic formula all known algorithms for its solution have non-polynomial complexity. But the com- Nk rv DÀ1 (k -+ +00) (6.1) plexity of an algorithm is determined by the number of elementary steps required to deal with a worst case, and in practice exponential algorithms often behave as where D is a constant. Several combinatorial enumeration problems can be reduced polynomial algorithms in many cases. In (24) the authors suggest the use of an in- to enumeration of walks in a graph (see (28), section 7.5). Thus the graph index dex, computable in polynomial time, which estimates in advance the complexity of 30 D. CVETKOviC AND P. RaWLINSON LARGEST EIGENVALUE OF A GRAPH 31 'i ~Ii a particular case of an N P-problem. (Then only an approximate solution would be (19) C. D. H. Cooper, On the maximum eigenvalue of a reducible non-negative real matrix, Math. Z. 131 (1973), 213-217. sought for a case of high complexity.) They ilustrate their ideas with a branch-and- (20) D. Cvetkovic, Graphs and their spectra, Univ. Beograd, Publ. Elektrotehn, Fak, Ser. Mat. Fiz 354 bound algorithm for the travelling salesman problem, that is, the problem of finding (1971), 1-50. a Hamiltonian path of minimal weight in a weighted graph. As one index of com- (21) D. Cvetkovic, Chromatic number and the spectrum of a graph, Publ. Inst. Math. (Beograd) 14(28) plexity they use the index of a spanning tree of minimal weight: the underlying idea (1972), 25-38. (22) D. Cvetkovic, A project for using computers in further development of graph theory, The Theory is that this index is a measure of branching, the least value being attained when the an Appl, of Graphs, Proc. 4th Internat. Cont held at Western Michigan University, Kalamaoo, tree is a path. Experimental results show a moderate to good correlation between MI, May 6-9, 1980, G. Chartrand, Y. Alavy, D. L. Goldsmith, L. Lesniak-Foster and D. R. Lick index and running time of the algorithm. (eds.), John Wiley and Sons, New York, 1981,285-296. liter- (23) D. Cvetkovic, Further expriences in computer aided research in graph theory, Graphs, Hyper- Finally we note that the index À1 of a graph has featured in the geographical graphs and Applications, Proc, Cont Graph Theory held in Eyba, October 198, H. Sachs (ed.), ature (15, 16, 48, 94, 95) in the context of traffc networks. 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(MR88:05128) ii in: Graphs and Other Combinatorial Topics, Proc. 3rd Czech. Symp, (Prague 1982), Teubner, Leip- (105) V. Vetchý, Estimation of the index of G2, Arh. Math. (Brno) 24 (1988), 123-136. n!~ zig, 1983, 223235. ~~: (75) P. Rowlinson, A deletion-contraction algorithm for the characteristic polynomial of a multigraph, Proc. Royal Soc, Edinburgh 10SA (1987), 153-160. 10 Contents 118 o. Introduction 4.3. A generalization of the divisor concept . . . . , , . , , . , . 118 4.4. Symmetry properties and divisors óf graphs. , , , , , , , . . 121 4.5. The fundamental lemma connecting the divisor and the spectrum 125 4.6. The divisor - an effective tool for factoring the characteristic polynomial. 128 4.7. The divisor - a mediator between structure and spectrum 131 4.8. . Miscellaneous results and problems...... , 134 5. The Spectrum and the Group of Automorphisms. 134 5.1. Symmetry and simple eigenvalues ...... 141 5.2. The spectrum and representations of the automorphism group 149 5.3. The front divisor induced by a subgroup of the automorphism group 153 5.4. Cospectral graphs with prescribed (distinct) automorphism groups 153 5.5. Miscellaneous results and problems...... 156 6. Characterization of Graphs by Means of Spectra . This introductory chapter is devoted mainly to the reader who is not familiar with 156 graph theory to help him to enter the topic of the book. The basic definitions and 6.1. Some families of non-isomorphic cospectral graphs 161 6.2. The characterization of a graph by its spectrum . 168 facts about the spectra of graphs are given together with a description of some 6.3. The characterization and other spectral properties of line graphs 178 general graph theoretic notions and necessary facts from matrix theory. For a general 6.4. MetricaUy regular graphs . , , . , . . . . ' . , . 183 introduction to graph theory the reader is referred to the books (BeCh2), (Bel' 1), 6.5. The (-1, 1, OJ-adjacency matrix and Seidel switching 185 6.6. :ßiiscellaneous results and problems. . , , , . , . . (Ber2), (Ber3), (BoMu), (Deo), (Har4), (Maye), (Nolt), (Sac9), (Wi,RJ2), (YHJIC), 189 (Xapa), a chemist may be especially interested in (Bala)t, and for a survey of matrix 7. Spectral Techniques in Graph Theory and Combinatorics theory we recommend the books (Gant), (MaMi). 189 7.1. The existence and the non-existence of certain combinatorial objects 193 7.2. Strongly regular graphs and distance-transitive graphs . . 199 7.3, Equiangular lines and two-graphs ...... , . . 203 0.1. What the spectrum of a graph is and how it is presented in this book 7.4. Connectedness and bipartiteness of certain graph products. 209 7.5. Determination of the number of walks . . , . 217 By a graph G = (qc, 'W) we mean a finite set qc (whose elements are called vertices) 7.6. Determination of the number of spanning trees 221 together with a set 'W of two-element subsets of qc (the elements of 'Ware called 7.7. Extremal problems. . , . . . . . 223 7.8. Miscellaneous results and problems. , edges). Similarly, a digraph (diTected graph) (qc, 'W) is defined to be a finite set qc 228 and a set ql of ordered pairs of elements of qc (these pairs are called directed edges 8. Applications in Chemistry and Physics 228 or arcs). The sets of vertices and edges are sometimes denoted by "f(G) and &'(G), 8.1. Hückels theory , . . , ...... 239 respectively. 8.2. Graphs related to benzenoid hydrocarbons, 245 If multiple undirected or directed edges are allowed, we shall speak of multigraphs 8.3. The dimeI' problem "..... 252 or multi-digraphs, respectively. These two cases include the possible existence of 8.4. Vibration of a membrane . . . , , 258 8,5, Miscellaneous results" and problems. loops (a loop is an edge or arc with both of its vertices identical). The terminology is 260 that of F. HAARY (Har4) except for the fact that in this book multi-(di-)graphs 9. Some Additional Results . . 261 are allowed to have loops. Although the term graph denotes what in many graph 9.1. Eigenvalues and imbeddings. 263 theoretical papers is called "a finite, undirected graph without loops or multiple 9,2. The distance polynomial . , 265 edges" (or, briefly, a "schlicht graph"), for the sake of readability we shall sometimes 9.3. The algebraic connectivity of a gmph , 266 of confusion) use the term graph in the most general meaning, 9.4. Integral graphs . . . . , , . . , 266 (when there is no danger 9.5. Some problems . , . . , . . ' . i.e., we shall mean undirected graphs, digraphs and even multigraphs and multi- 268 digraphs. Appendix. Tables of Graph Spectra. 324 Two vertices are called adjacent if they are connected by an edge (arc). The ad- Bibliograiihy. . . jacency matrix A of a multi-(di-)graph G whose vertEjx set is jXi, X2, ..., xnJ is a square 360 Index of Symbols. matrix of order n, whose entry aj.j at the place (i, j) is equal to the number of edges 361 Index of Names 364 t Recently two books on Hückel theory have appeared (see the end of p, 359). Subject Index . ~ ~ :;~ .. '" 0+ -+ ¡g :: s ~ t. ~. ,. s:; ~. 8. e: tt tt q: ~ f!. ;; ~ ~ fñ' ~ tt fñ' '- ~. CJ i: Cl ~ ,. S~P: ~ S co Q' 0 ~ II g 'd 16 0 ~ ~ t- 8 ~ ~ co co t- ~ Il~ ~ g- § co ~ c: ~ ~ ~ r- - co co g- ~ ,.~~ ~~:3. II g-~~~,.:; 0 ~ c: ~ ~ g-¡.~p.CJ¡:COCO""~ co .. tj Ci s' $ Q' ~ g- co -; g-c: ::. ~c: c: i: S ~o" ,. ::CD co ~§::¿P -= S ~S ~ o .. "' :; ~ 8. i: ~ co ~ ¡; go g- 0 ~ ~ ~ ~ ~ p. co Ci ~ ~ co p. ~ g. ? ~ ~ ~ .. ~ g ~ s' ~ ~ :; ~ II ~ I- tj .. . ~ ,. ~ Q' ~ p. 0 ~ ~ ~,: i: UJ :: r- ~ ~ ~ ... t: ~-= o ~ c1' S _. t: o §. g. tt ~ II "d ~ tj 0+ ft ~ 13' "d t= ~:; OQ ,. gg~Sott""~~S:=: ~Otj..cotj~f~ UJ. UJ S' co ... 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Q' -- ~ co ¡: .."'... -= ... c: ,. ~ _.. . co p. :; g co,. i: p. ~ ~~~ S -= ~ tj S 0" Ci i: '" of p. ~ ~ ~ e: ~. 0 i: 15 ~ ~ ~ UJ ~ ~ " CD .g,. ~p. ~UJ 8 ~,. o i: 0 i: ¡:e: '"... ¡g. "d~ UJ 0 c::; '"15 ~ 0"~ p:, g -8 ~0 UJc: ::~ UJo...co .... g tj:: '" ~ ,. o ~ co ~ ~ co UJ ~ 1J~ tj":tj-';....,. co tj~ s.. 0.B ~ Q. ~g"g :; co ;; ~ c: c: co '2 -= tj tj ~ co c; 0+ ;- c: UJ ~ ~ f!. c::; ~ ¡. "d c: co 0 0 "d S .. co ¡g § ~. ~ q: i: ,. f' ~ 8: ~ . ~ ~ êS~' UJ t- ~ - OQ =i -,. tj.. Ci ,,. "d c: 0" ~ ,. tj ~. ,. S' co co i= i: ~ co &r:: 0 gi ~ Q' ~ ¡: 6: 'p. gi -- æ. ~ co p. õ ¡g co II q 0 co S ,. "d co UJ~' tj ...,. II ,. ~ ~,.oco~ c: ~ ~ ~ b: co &..§ tj .,. co ~ 0" i: _. t: ~ ~ OQ ~::. 9. '" o~~~s: co ,. ~ ¡: ,. I- ~ c1' ,. ~ !f UJ ~ - ~ ~ "" ~- tjq~q~ o.' S ,:'" CD " .. t: ~,. i: co ... 0 .. .¿ co ~ .. tj '" -= ,. CO..c: :: E; co " 0 '" p ...? co tj ~ .g- f!. 8 ~ "' tj p. ~ OQ H co ",' ¡: UJ co .. ft !; "". 0" ~ co -= :; ~ S 0 'õ g: 15 'g' tj -;.. ~ S: tj ~ c: ~ o e:;~ -i ,. 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b: ;; tt ~ S tt ~ g- ~ ~ ~ S' S' g ttco cott i: :; (o: tt co ~ i:CJ 8. "d... ~ ..:i co..tt tico tCl ttCl f!.g. i:co t- o c:. S' (E ..co' ~ ~ t:: ClH ,. "d co W ,. ~ S' ~ t- UJ co ~ I- UJ co UJ ... gi ~ 8 ~ UJ:j "drÐ;: t- g c: Q, -= gi õ1 ~ CJ ~¡: .!""' !2 §"1 ~ co rn a"d ~ OOQ,.. CJ i:~ co o"d~ :3. ~ O"g.i: UJ UJco!f co ,. co~ ,. 0 co,. t- CO~~i:''=ie:tj giUJ Op.~""~~ i: ~ UJ~~COOUJtjO ,. UJ UJ gi ~ :: tj8o""otjco ¡: UJ 0 p. ~ Q' i: '" co a ~ ~ ~ ~ co ¡: ~,. "d ¡g S' § ~ gi e: =i ~ '" CJ,. ~ gi "g: ~ ~ ~ ¡:s'"g _. g.tr coS ~;: p.~ p.g. 0" o. :-. ,. c:CJ 2,. ttco 12 e: S' ~ 0 §. ~ p. ~ i: p... ~ § a ~ ~ 8 B s "d i: Q. ;: !f tt p. g- 0" ~ ? gi ;; co ~ -= fñ' g:~~ Ci g,e:aa'g~ t.~"¡:;CJ g-¡: S :: got7c: "d co p tj..t: ... ttCJ G .. ~~ ~,..::.. N 'O.."d ~ gi ,. ~'",. ê. 0 II q: ~ : ~. ~ ~. g ~ ~ ~ S l ~,~ S- ~. ~ ~ s. :: ~ ~ ,. co ~ c: gi ~ ~ 0 "d i: ~ ~ ~ o ~i:~co~pcoco ,.~UJ"d..tjO i-.. 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UJ :3. g- c:::::¡:~UJUJ-=tjco¡:""~ fñ' tj ~ tj co' ~ :: ~, ~ l~ I CD" ~COI'"" I's:.' o:o~~~sÕ'8 j Ci g. g. g: a ~ ~ S S' S ~ ~ 6 e: ~ 0+,. ~ co g. S' ~ S ~ '" tj gi t: UJ. ,. ;: ~Otl gi gi CJ S :; g ,.q ~ q :: S .. '0 UJi:coll~ ~ S' ~ ~ i:coi:õ"" tj g ;: CJ ~,.i:~8~Oco ~ i: i: ~ p:, s ¡;~ u co..~ UJ~CO~~'"i~ L. o ci ;- tr õ fñ' g- :; :3 :. tt 0" co ;. fñ' '" 00 co~ O"d~tj~ CD ;: ,. ~ "";:g.o !5~~ Sooco~o~ 1 ~ ~~. ~ § l':'co ~ §:: ~ ~ ~ ,. .. :; (J ~p.UJto ..~i: p."dtj;: ,. ~ UJ _. p. t= .. ,. ~ fñ' ,. .. :- ¡i g' ,. i:gi ,.0" 00!fCO :; UJ p: ~ ,. UJ¡:~ ¡: :; S',. ~ i:8g.,. ;:;:~ ~cogogico:; ~ s' p. ;: UJ 2 ~ ä. ~ ~ 0 t" t7 i: 0 :: ~ c: ... ~ cr"" co c: tj ~ ~ co "d0~ "dl'S":3 ~::,.o~ f(;: $' ~ ~ ~ ~ l ~ ~ ~ ~ ~ ~ g- ~. $ o..~ l5 g- ~~..~~ ;:. c: ;: UJ1" M t:~..0 gi g. "d. :3~"d .. g- ..~. ttco. e: ~ S coPco~~~ co g ~ :; ¡: ~tjcoO S "E '"~:;c:§ UJ tr 8- g-ttq:"¡coUJ e: t: ~ E o' ~s.. ~""::-= ~CJo :;crtj"dc: ~;:ai: ~ en ~.""UJs; i- ;S. ""OONui¡: pq ~ \-. cp Q .. o'~ ~co i: 0 coUJ ,."" ~ ~ i:~ ... -=0 ¡:¡: ~:: ¡:~ ~15 (J ~ '; t=a. ~::g- go~~:- 8 ¡ga ~§ a g,~ g.~ ~(J&f1 g,g ~ 9 0.2. Some more graph theoretic notions and conventions 15 14 O. Introduction If there is an arc from vertex x to vertex y, we shall sometimes indicate this by is of interest, especially in some cases when the spectra of compound graphs can be writing y' x; x and yare neighbours of each other, x is a rem' neighbour of y and y expressed in terms of the spectra of simpler graphs. is a front neighbour of x. A cycle of length n, denoted by Õn, is a digraph with the vertex set ¡xi, ..., :rnl The plan of the book is as follows: having arcs (Xi, x¡+1), i = 1, ..., n - 1, and (xm Xl)' A linear directed graph is a di- In Section 0.2 some general graph theoretic notions are given together with some graph in which each indegree and each outdegree is equal to 1 , i.e., it consists of cycles. conventions used throughout this book. Section 0.3 contains the necessary theorems A spanning linear subgraph of a multi-(di-)graph G, i.e., a linear subgraph of G from matrix theory and describes some basic facts about graph spectra. which contains all vertices of G is sometimes called a linear factor of G. A linear In Chapters 1, 3, 4, 5, 6 relations between spectral and structural properties of factor of a multi graph consists of disjoint copies of K2. graphs are described. A regular spanning sub graph of degree s of a multi graph G is called a (regular) Chapter 2 describes the relations between the spectrum of a graph constructed by factor of degree s or, briefly, an s-factm' of G. operations on some given graphs and the spectra of these graphs themselves. I,n a multi-(di-)graph any sequence of consecutive edges (arcs) (having in mind Chapters 7 and 8 describe the applications of the theory developed in Chapters 1 the orientation in directed case) is called a w(ilk. The length of the wa.lk is the number to 6. Chapter 7 is related to the applications in graph theory and combinatorics and of edges (arcs) in it. A walk can pass through the same edge (arc) -more than once. Chapter 8 gives the applications beyond mathematics, i.e. in chemistry and physics. A path of length n - 1 (n ~ 2), denoted by Pn, a is graph with n vertices, say Chapter 9 contains some additional material which did not fit into the clàssification Xl' ..., Xn, and with n - 1 edges in which Xi and Xi+1 are connected by an edge for of the other chapters. The Appendix contains numerical data on graph spectra and i = 1, ..., n - i. the corresponding characteristic polynomials. A multi-(di-)graph is (strongly) connected if any two of its vertices are joined by a The last section in each of Chapters 1 - 8 has the title Miscellaneous results and path (walk). A multigraph is disconnected if it is not connected, and it then consists problems. At these places some additional material is reviewed, partly in form of of two or more parts called components, two vertices being in different components exercises and problems. Section 9.5 gives a list of unsolved problems. if they cannot be joined by a path. A vertex ;¡ is called a mdpoint and an edge u is The Bibliography contains more than 650 references from both the mathematical literature. Although the authors believe that all important papers called a bridge if the deletion of x or u, respectively, ca,uses an increase of the num- and the chemical ber of components. from the viewpoint of this book are included, they are aware that a complete biblio- The length of a shortest path between two vertices is called the distance between graphy on graph spectra is almost impossible to compile because of the thousands of the vertices. The diameter of a connected multigraph is the largest distance between chemical papers where graph spectra are only mentioned in passing and the hundreds the vertices in it. of papers on association schemes, block designs, and related combinatorial objects A circuit Cn of length n is a regular connected graph of degree 2 on n vertices. Con- where eigenvalues are also involved, although not always in an important way. sidered as a subgraph, Cl is a loop, C2 is a pair of parallel edges, C3 is atriangle, C4 is a quadrangle. The girth of a multi-(di-)graph is the length of a shortest circuit 0.2. Some more graph theoretic notions and conventions (cycle) contained in it. A multigraph G is said to be properly coloured if each vertex is coloured so that . We shall now give some more defintions of the graph theoretic notions frequently adjacent vertices have different colours. G is k-colourable if it can be properly used throughout the book. We shall also point out some standard notations and coloured by k colours. The chromatic number X(G) is k if G is k-colourable and not explain some conventions used in the subsequent text. (k _ l)-colourable. G is. called bipartite if its chromatic number is 1 or 2. The vertex A graph H = (i!, 1/) is said to be a subgraph of the graph G = (2l, 0l) if i! c 2l set of a bipartite multigraph G can be partitioned into two parts, say 2l and i!, in and 1/ cOl. The graph H is called a spanning subgraph or a partial graph of G if such a way that every edge of G connects a vertex from 2l with a vertex from i!. i! = 2l. If 1/ consists of all the edges from 0l which connect the vertices from i!, If 0l denotes the edge set of G, we have also the following notation: G = (2l, i!; qt). then H is called an induced subgraph. An induced subgraph is said to be spanned by If G is connected and has an edge, then 2l and i! are non-void and (up to an intei:- its vertices, and a partial graph is sometimes said to be spanned by its edges. change) uniquely determined. If the vertices are labelled so that The number of edges incident with a vertex in an undirected graph is called the 2l = ¡xu X2, ..., xm), i! = ¡xm+1' Xm+2, ..., xin+n)' degree or the valency of the vertex. Note that an undirected loop is counted twice, thus its contribution to the valency of the vertex to which it is attached is equal then the adjacency matrix of G takes the form to 2. If all the vertices have the same valency r, the graph. is called 'regular of degree r. A = (0 BT) In digraphs we shall distinguish between the indegree or rear valency and the B 0 ' outdegree or front valency (of a vertex) by indicating how many arcs go into and go where.B is an n X m matrix and BT is the transpose of B. out from the vertex, respectively: 16 O. Introduction 0.3. Some theorems from matrix theory 17 A multigraph is called semiregitlar of degrees ri, 1'2 (possibly ri = 1'2) if it is bi- partite, each vertex has valency ri or 1'2' and each edge connects a vertex of matrix where valency ri with a vertex of valency 1'2' Vii = 1 if ui issues from Xi' K" denotes the complete graph on n vertices (any two distinct vertices of K" are connected by an edge). Km,,, is a complete bipartite graph on n + m vertices; Ki." is Vii = -1 if Uj terminates in Xi' called a star. The complete k-partite graph on ni + n2 + ... + nk vertices is denoted Vii = 0 otherwise. by K"lO "...... "k' A forest is a graph without circuits, a tree is a connected forest. In the majority of cases we shall use the following standard notation. The complement G of a graph G is the graph with the same vertex set as G, where The number of vertices of a graph is denoted by n, the number of edges or arcs any two distinct vertices are adjacent if and only if they are non-adjacent in G. by m. The degree of a regular graph is denoted by l' as is the index of a graph (see Obviously, G = G. A graph without any edges is called totally disconnected, its com- the next section). The symbol I means a unit matrix in general and 111 is a unit matrix plement is a complete graph. of order n. The symbol J denotes a square matrix all of whose entries are equal to i. The subdivi'ion graph S(G) of a graph G is obtained from G by replacing each of The transpose of a matrix X is denoted by XT, and rkX is the rank of X. its edges by a path of length 2, or, equivalently, by inserting an additional vertex The Kronecker symbol Oii is defined by Oii = 1 and Oii = 0 if i =+ i. into each edge of G. Clearly, S(G) is a bipartite graph (~, Il; Ol) where ~ and Il a I b means a divides b. are the sets of the original and of the additional vertices, respectively. In the case of undirected multigraphs the spectrum consists of real numbers. In The line graph L(G) of a graph G is the graph whose vertices correspond to the that case, the eigenvalues )'i, À2, ..., À" are ordered so that always Ài = l' ~ )'2 edges of G with two vertices being adjacent if and only if the corresponding edges ~ ... ~ À". in G have a vertex in common. Other notations and graph theoretic concepts wil be given at the place of their The vertex-edge incidence matrix R of a loopless multigraph G = (~, 0//) is defined use. as follows: Let

~ = rxi, X2, ..., x,,l, Ol = rui, 1L2, ..., uml. 0.3. Some theorems from matrix theory and their application to the spectrum of a graph R = (bij) is an n X m matrix where bij = 1 if Xi is incident with (i.e., is an end vertex of) Uj, and bii = 0 otherwise. The edge-vertex incidence matrix is the transpose RT Some fundamental properties of spectra of graphs (or, more generally, multi-digraphs) ofR. can be established immediately by using several theorems of matrix theory. We shall The adjacency matrix of a multi-(di-)graph G is denoted by A = A(G). The formulate in this section only the most important matrix theorems. Others, which v(tlency or degree matrix D of a multigraph is a diagonal matrix ~ith the valency Vi are also useful, wil be given in the subsequent chapters as lemmas at the places of vertex Xi in the position (i, i). where they are needed. It is not difficult to see that, for a graph G, the vertex-edge incidence matrix R, The set of eigenvectors belonging to an eigenvalue À along with the zero vector the degree matrix D, and the adjacency matrices of G, L(G), and S(G) are connected forms the eigenspace belonging to À. The geometric multiplicity of (tn eigenvalue À is the by the following formulas: dimension of its eigenspace. The algebraic multiplicity of À is the multiplicity of À considered as a zero of the corresponding characteristic polynomiaL. The geometric A(G) = RRT - D, multiplicity is never greater than the algebraic multiplicity. A matrix X is called symmetric if XT = X. A(L(G)) = RTR - 21, Theorem 0.1 (see, for example, (MaMi), p. 64): The geometric and algebraic multi- A(S(G)) = (~ ~T). plicities of an eigenvalue of a symmetric matrix are equal. In the subsequent text the multiplicity of an eigenvalue wil always mean the alge- The above definitions and formulas can easily be generalized for arbitrary multi- braic multiplicity. graphs. A matrix is called non-negative if all its elements are non-negative numbers. The (0,1, -1)-incidence matrix V of a loopless multi-digraph G with vertices Since the adjacency matrix of a multi-(di-)graph G is non-negative, the spectrum Xl' Xi, ..., Xli and arcs ui, ui, ..., Um is defined as follows: V = (Vii) is an n X m of G has the properties of the spectrum of non-negative matrices. For non-negative matrices the following theorem holds.

2 CvetkoviclDoob/Sachs 0.3. Some theorems from matrix theory 19 18 O. Introduction

Theorem 0.2 (see, for example, (Gant), voL. II, p. 66): A non-negative matrix If the adjacency matrix is symmetric, the converse of the last statement also holds, always has a non-negative eigenvalue l' such that the moduli of all its eigenvalues do not as shown by the following theorem. exceed r. To this "maximàl" eigenvalue there corresponds an eigenvector with non- Theorem 0.4 (see, for example, (Gant), voL. II, p. 79): If the "maximal" eigenvalue neg(itive coordinates. l' of a non-negative matrix A is simple and if positive eigenvectors belong to l' both in A ltid AT, then A is irreducible. In the subsequent text a vector with positive (non-negative) coordinates wil be called a positive (non-negative) vector. A matrix A is called reducible if there is a Theorem 0.5 (see, for example, (Gant), voL. II, p. 78): To the "maximal" eigenvalue r of a non-negative matrix A there belongs a positive eigenvector both in A and AT if permutation matrix P such that the matrix P-IAP is of the form (X 0), where and only if A can be represented by ii permutation of rows iind by the siime permutation X and Z are square matrices. Otherwise, A is called irreducible. Y Z of columns in quasi-diagonal form A = diag (A¡, ..., As), where A¡, ..., As iire irredu- Spectral properties of irreducible non-negative matrices are described by the cible matrices each of which hiis l' a,s its "maximal" eigenvalue. following theorem of FROBENIUS. We shall now list some more theorems from the theory of matrices showing Theorem 0.3 (see, for example, (Gant), voL. II, pp. 53-54): An irreducible non- new spectral properties of graph. negative miitrix A iilwiiys has a positive eigenvalue l' that is a simple root of the chamc- teristic polynomial. The modulus of any other eigenvalue does not exceed r. To the Theorem 0.6 (see, for example, (Gant), voL. II, p. 69): The "maximiil" eigenvalue "miiximal" eigenviilue l' there corresponds a positive eigenvector. Moreover, if A has h 1" of every principal submatrix (of order less than n) of ii non-iiegiitive matrix A (of eigenvalues of modulus 1', then these numbers are all distinct and are roots of the equation order n) does not exceed the "maximal" eigenvalue l' of A. If A is irreducible, then Àh _ rh = O. More generally: the whole spectrum (À¡ = 1', ..2, ..., Àn) of A, regarded as ii 1" ~ l' iilwiiys holds. If A is reducible, then 1" = l' holds for at least one prin¡;piil sub- system of points in the complex À-plane, is mapped O'ìto itself under a rotation of the matrix. plane by the angle 2n. If h ? 1, then by a permutation of rows and the same permutation Theorem 0.7 (see, for example, (CoSi 1)): The increase of any element of a non- h negative matrix A does not decrease the "maximal" eigenvalue. The "rnaximal" eigen- of columns A can be put into the following "cyclic" form val'ue increases strictly if A is an irreducible matrix.

0 A¡2 0 ... 0 ) Theorems 0.6 and 0.7 state that in a (strongly) connected multi-(di-)graph G 0 0 An .. . 0 every subgraph has the index smaller than the index of G. A -I: i. (0.1) Theorem 0.8 (see, for example, (MaMi), p. 64): All the eigenvalues of a Hermitiant 0 0 0 .. . Ah-i.h matrix are 1'eal numbers. Ah¡ 0 0 ... 0 ) Theorem 0.9 (see, for example, (Hof1)): Let A be (i real symmetric matrix whose greatest and smallest eigenvalues a-re denoted by l' and q, respectively. Let æ be the eigen- where there are square blocks. along the main diagonal. vector belonging to r. For a principal submatrix B of A, letti' be the smallest eigenvalue If h? 1, the matrix A is called imprimitive and h is the index of imprimitivity. whose eigenvector is denoted by y. Then q' ~ q. If q/ = q, vector y is orthogonal to the Otherwise, A is primitive. projection of vector æ on the subspiice corresponding to B. a Hennitian matrix with According to Theorem 0.3, the spectrum of a multi-(di-)graph G lies in the circle Theorem 0.10 (see, for example, (MaMi), p.119): Let A be eigenvalues ),¡, ..., Àn and B be one of its principalsubmatrices; let B have eigenvalues 1..1 ~ 1', where l' is the greatest real eigenvalue. This eigenvalue is called the index of G. The algebraic multiplicity of the index can be greater than 1 and there exists fL¡, ..., fLm' Then the inequalities Àn-m+i ~ fLi ~ Ài (i = 1, ..., m) hold. a corresponding eigenvector which is non-negative. These inequalities are known as Oauchy's inequalitie; and the whole theorem is Irreducibility of the adjacency matrix of a- graph is related to the property of also known as interlacing theorem. connectedness. A strongly connected multi-digraph has an irreducible adjacency matrix and a multi-digraph with irreducible adjacency matrix has the property of strong con- Theorem 0.11 (C. C. SIMS, see (HeHi))H: Let A be a real symmetric matrix with nectedness (DuMe), (Sed 1). In undirected multigraphs the strong connectedness eigenvalues J.1, ..., Àn- Given a partition p,..., n) = Ll¡u L12 U ... u Llm with ILlil =ni? 0, reduces to the property of connectedness. t The complex matrix A = (aij) is called Hermitian if AT = A, i.e. aji = aij. According to Theorem 0.3, the index of a st1'ngly connected multi-digraph is a H Recently W. H. HAEMERS (Haem) has shown that the interlaoing properties also hold for simple eigenvalue of the adjacency matrix and a positive eigenvector belongs to it. matrices A and B of this theorem.

2* 0.3. Some theorems from matrix theory 21 20 O. Introduction A square matrix with the property that its minimal and characteristic poly- consider the corresponding blocking A = (Aij), so that Aii is an ni X n¡ block. Let eii nomials are identical is called non-derogatory. Thus Proposition (e) says that a square be the sum of the entries in Aii and put B = (eii/ni) (i.e., ei¡/ni is an average row sum matrix which has all eigenvalues distinct is non-derogatory. in Aii). Then the spectrum of B is contained in the segment (Àm Àd. We shall now describe some more basic properties of the spectrum of an undirected If we assume that in each block Ai¡ from Theorem 0.11 all row sums are equal, multigraph. The facts wil be given almost without an.y proof for the convenience then we can say more. of the reader. The proofs can be found at the corresponding places in the subsequent Theorem 0.12 (E. V. HAYNSWORTH (Hayn); M. PETERSDORF, H. SACHS (PeS1)t): chapters. Let A be any matrix partitioned into blocks as in Theorem 0.11. Let the block Aii have The adjacency matrix of an undirected multigraph G is symmetric (and, therefore, real numbers, according to Theorem constant row sums bii and let B = (bii). Then the spectrum of B is contained in the Hermitian) and the spectrum ofG, containing only spectrum of A. (having in view also the multiplicities of the eigenvalues). 0.8 lies in the segment ( -1', 1'). Let (À1, ..., À,.) be the spectrum of a multigraph. Twice the number of loops is equal The square matrices A and B are called similar if there is a (non-singular) square to the tmce of the adjacency matrix. Therefore, we have for multigraphs without loops matrix X transforming A into B, i.e., such that X-lAX = B. Each symmetric matrix tr A = 0, i.e., )'1 + ... + Àn = O. The number of vertices is, of course, equal to n, and each matrix which has all distinct eigenvalues is similar to a diagonal matrix. and for undirected gmphs without loops or multiple edges the number m of edges is given by If A is the adjacency matrix of a multigraph, then A is symmetric and, consequently, similar to a diagonal matrix D, namely, D = (ÖiiÀi)' m = ~ i: ).7 (see Section 3.2). 2 i=1 We mention the famous Oayley-Hamilton Theorem which says that each square It is stated in (CoSi1) that for the index r of a connected graph the inequality matrix A,¡satisfies its own characteristic equation, i.e.: 2 cos ~1 ~ r ~ n - 1 holds. The lower bound is attained by a path, and the upper If f(À) = IÀl - AI, then f(A) = O. n+ bound by a complete graph. If we omit the assumption of connectedness, then for The minimal polynomia,l m(À) of A is the polynomial m(J,) = ÀI' + ... such a graph without edges we have l' = 0 and otherwise r ;S 1. that For the smallest eigenvalue q of the spectrum of a graph G the inequality (i) m(A) = 0, -1' ~ q ~ 0 holds. For the graph without edges we have q = O. Otherwise q ~ -1. (ii) under condition (i), the degree tt of m(À) has its minimum value. This is a consequence of Theorem 0.9, since the subgraph K2 corresponds to a prin- cipal submatrix with least eigenvalue equal to -1. We have q = -1 if and only if Then the following propositions hold: all components of G are complete graphs (Theorem 6.4). The lower bound q = -1' is uniquely determined by A. (a) m(À) is achieved if a component of G having the greatest index is a bipartite graph (Theo- (b) If F(À) is any polynomial with F(A) = 0, then m(À) I F(À); in particular, rem 3.4). According to the foregoing, the following theorem describes the fundamen- m(À) I f(À). tal spectral properties of (undirected) graphs. (c) Let rÀ(ll, ),(2),..., À(k)) be the set of distinct eigenvalues of- A, À(') having algebraic multiplicity m.. Then Theorem 0.13: For the spectmm(Ài, ..., Àn) of an (undirected) graph G the following statements hold: f(À) = (À - ),(l))mi (À - À(2))m, ... (À - À(k))'n~ and 1 ° The numbers )'i, ..., Àn are real and Ài + ... + ),11 = O. m(J,) = (ì, - ì,(l))ai (À - À(2))a, ... (À - À(k))a. 20 If G contains no edges, we have Ài = ... = Àn = O. 30 If G contains at least one edge, we have where the q. satisfy o ~ q. ~ m. (x = 1, 2, . ", k). 1~r~n-1, (0.2) (d) If ~4 is similar to a diagonal matrrx, then all q. are equal to 1: -1' ~ q ~ -1. (0.3) ~(À) = (J, - À(l)) (À - À(2)) ... (À - À(k)). In (0.2) the upper bound is attained if and only if G isa complete graph, while the lower (e) Let A have order n. If A has all distinct eigenvalues, then bound is reached if and only if the components of G consists of gmphs K2 and possibly Ki. m(À) = f(À) = (À - ì/1)) (ì, - ì,(2)) ... (J, - À(n)). In (0.3) the uppe1' bound is reached if and only if the components of q are complete graphs, and the lower bound if and only if a component of G having the greatest index is t See Theorem 4.7. 22 O. Introduction i 1. Basic Properties of the Specirum of a Graph ¡ a bipartite graph. If G is connected, the lower bound in (0.2) is replaced with 2 cos ~ . Then equality holds if and only if G is a path. n + i We shall now list some spectral properties of regular multigraphs. The index is equal to the degree (CoSi i). It can easily be seen that this holds for disconnected multigraphs tob, but then the index is not a simple eigenvalue. The multiplicity of the index is equal to the number of components. It can be seen immediately that the vector having all coordinates equal to i is an eigenvector that corresponds to the index. The eigenvectors of the other eigenvalues are orthogonal to this vector, i.e., the sum of their coordinates is equal to O. Further spectral properties of graphs can be obtained using the fact that the coefficients of the characteristic polynomial are integers. It follows from this that the The ordinàry spectrum of a (multi-di- )graph G is the spectrum of its adjacency elementary symmetric functions and sums of k-th powers (k a natural number) of matrix, but there are various other methods of connecting a spectrum or a cha- eigenvalues are integers, too. Since the coefficient of the, highest power term of the racteristic polynomial with G. A general method of defining characteristic poly- characteristic,polynomial is equal to i, rationnl eigenvalues (if they exist) are integers. nomials (in one or more variables) and graph spectra is outlined, the most important spectra currently used and their interrelations are discussed, and it is shown how the coefficients of the corresponding characteristic polynomials can be obtained directly from the "cyclic structure" or from the "tree structure" of G, respectively. Eventually, the generating function for the numbers of walks of length k (k -' i, 2, ...) in G is expressed in terms of the ordinary characteristic polynomial and some conclusions are drawn.

1.1. The adjacency matrix and the (ordiary) spectrum of a graph

In order to obtain an arithmetic method for describing and investigating the structural properties of a finite (directed or undirected) (multi-)graph G, it seems quite reasonable to start with the adjacency matrix A of G. Obviously, G is uniquely determined by A, but the converse statement does not, in general, hold true since the ordering (numbering) of the vertices of G is arbitrary: To each graph G there corresponds uniquely a class d = d( G) of adjacency matrices, two adjacency matrices A and A* belonging to the same class (i.e., determining the same graph) if and only if there is a permutation matrix P such that A* = P-IAP. Thus the theory of graphs G may be identified with the theory of these matrix classes d and their invariants. An important invariant of a class d is the charac- teristic polynomial PGV..) = 1J. - AI with A E d(G), or, what amounts to the same thing, the spectrum Sp(G) = P'l, Â2, ..., Ân), where the )..;'s are the roots of the equa- tion PG(Â) = 0 (i.e., the eigenvalues of A).t The main question arising is this: how much information concerning the structure of G is contained in its spectrum, and how can this information be retrieved from the

spectrum ~ Of course, the amount of information 'contained in the spectrum must \

t In order to avoid confusion, this "ordinary" spectrum wil later sometimes be called the P-spectrum of G, and it wil be denoted by Spp(G). 1.2. A general method for defining different kinds 25 24 1. Basic properties of the spectrum of a graph not be overestimated, since the spectrum remains invariant not only under the group linear equatiDns of permutations, but also under the group of all orthogonal (and even of all non- ÀXi = L xi; (i E gr),t (1.1) singular) transformations: Thus the spectrum reflects common properties of all k- those graphs the adjacency matrices of which may be transformed into one another graph, the multiplicity ail; by some non-singular matrix. Any such matrix transforming the adjacency matrix A the value of À being suitably chosen; if G is a multi-( di- ) of the adjacency k . i is to be taken into accDunt by considering Xl; exactly ail; times of a graph G into the adjacency matrix AI of some graph GI not isomorphic with G as a member of the right side sum of (1.1). Obviously, (1.1) may be given the shorter is subject to stringent diophantine conditions as all entries of A and AI are required fDrm to be non-negative integers: Therefore it may be expected that the classes of iso- Àæ = Aæ, ( 1.2) spectralt graphs are, in a sense, not too extensive. Jsomorphic graphs are, of course, isospectral, and it has been conjectured, conversely, that any two isospectral graphs A = (ail;) being the adjacency matrix Df G and æ denoting a column vectDr with are isomorphic; however, this is not true. It is very easy indeed to find isospectral compDnents Xl; (k E gr). As a necessary and sufficient condition for the existence of non-isomorphic digraphs, e.g., all digraphs with n vertices, containing no cycle, a nDn-trivial sDlution of (1.1) or (1.2), we have have the same spectrum (0, 0, ..., 0) (see 1.4, Theorem 1.2). An essentially different situation arises if only undirected (multi-)graphs are IÀl- AI = PG(À) = 0, taken into consideration, and the construction of pairs of isospectral non-iso- i.e., the possible proportionality factors À are identical with the eigenvalues of G. morphic (multi- )graphs becomes more and more difficult if one passes from multi- This way of reasoning has the advantage of being particularly intuitive, as the graphs to graphs and from graphs to regular graphs. Thus the spectral method I components of an eigenvector may be directly interpreted as "weights" of the cor- may be expected to be particularly efficient, when applied to the class of regular I I responding vertices; at a later stage we shall find that the immediate rationale of the graphs. spectrum (via equations (1.1)) by inspection of the graph itself and, particularly, Nevertheless, in the theory of block designs it has been shown that, even among simultaneous consideration of its eigenvectors, wil be very useful for a series of strongly regular graphs (which form a narrow subclass within the class of all regular investigations and proofs. graphs) with sufficiently many vertices, pairs of isospectralnon-isomorphic graphsH Certain applications necessitate the determination of the weights xt of the ver- are in fact not uncommon; see Chapter 6. tices in such a way that xt is proportional not to the sum (as above) but to the mean This phenomenon may, on the one hand, be taken as an indication of the scope value of all those xt corresponding to the (front) neighbours of i, i.e., the xZ are and the bounds of this special spectral method; on the other hand, it probably required to satisfy the system of equations reflects a peculiarity of the theory of block designs, showing that there are indeed

close relations between this theory and the spectral method. 1 ÀXi = - LXi; (iE ¿().H (1.3) di l;,i

(1.3) may be replaced by 1.2. A general method for defing diferent kids of graph spectra ),Dæ = Aæ, (1.4) In this section we shall consider another very natural approach to the spectral yielding immediately method which, by appropriate variation, yields arbitrarily many different "spectra", i.e., systems of numerical invariants. I),D - AI = 0 Let us start with the ordinary spectrurn Spp(G), as an example. We consider a set as a necessary and sufficient condition for the existence of a non-trivial solution of of n (unspecified) variables Xi; being in (1, l)-correspondence with the set of vertices k (1.3) or (1.4). Thus we are led to introduce as a modified characteristic polynomial (k = 1,2, ..., n) of a given (multi-di-)graph G = (gr, 0/). We try to find numerical 1 values xi for all of the X,1c not all equal to zero and such that for each vertex i the QG(À) = iD IÀD - AI = À" + qiÀ,,-i + ... + qn (1.5) corresponding number x? is proportional to the sum s? of all those xi corresponding to the (front) neighbours of i (i.e., such that the ratio s?;x? is the same for all i). In other words, the xi are to satisfy, in a nDn-trivial way, the system of hDmogeneous t k . i means that k is a (front) neighbour of i (and i is a (rear) neighbour of k). tt di here denotes the (out- )degree or (front) valency of vertex i, i.e. the number of arcs issuing t Graphs having the same spectrum are called isospectral or cospectral. 't Such a Pair of Isospectral Non-isomorphic Graphs is sometimes given the acronym PING; from vertex i; it is assumed here that di ? 0; the diagonal matrix D = (Oikd¡) is called the more information about the construction of PINGs wil be found in Chapter 6. (out-)degree or (fmnt) valency matrix of G. 27 26 1. Basic properties of the spectrum of a graph 1.2. A general method for defining different kinds with corresponding spectrum corresponding to 8 are

ÂXi = E SiT.XT. (i E f!), (1.2) 8pQ(G) = P'I, ,12, ..., )'n)Q' (1.6) kEgl Note that 8G(Â) = IU ~ 81 = IU - J + 1 + 2AI

= )," + SI.1,,-1 + .,. + S", ( 1.3) QG(Â) = IU - D-IAI = ¡U - AD-II. (1.5)' 1 1 1 8Ps(G) = PI' ,12' ..., )'n)S, (1.4) Let D2 . (biT. Vd;) and A* = D2 (D-IA)D-2. Then respectively. In this connection, two more spectra derived from the matrix of admittancet, A* = D-f AD-f = (V::~l,, C = D - A, should be mentioned. Some authors (W. N. ANDERSON Jr., T. D. MORLEY (AnMo); M. FIEDLER (Fie 1)) consider the polynomial QG(Â) = IU - A*I. (1.5)" GG(Â) = IU - ci = !U- D + AI = )," + CiÂ"-1 + ... + c" (1.15) For an undirected multigraph G, A* is symmetric and, consequently, 8pQ(G) is reaL. with corresponding spectrum In (1.5) D appears in a multiplicative manner; D may also be introduced in an 8po(G) = (.11, ,12' ..., ,1,,)0 (1.6) additive way: starting from (using, of course, different notation); A. K. KEL'MANS (KeJI 1) introduces a poly- nomial ~=~+E~=E~+~ (i E f!) ( 1.7) k~ ~i B1(G) = ~ 1,11 + ci ( 1. 7) we obtain another characteristic polynomial ,1 of order n - 1; clearly, RG(Â) = IU - D - AI = ,1" + r1Â"-1 + ... + rn (1.8) . B1(G) =( -1)" - GG(-)')' with corresponding spectrum ,1 so that no special symbol is required for the Kel'mans spectrum. 8PR(G) = (,11' )'2' ..., Ân)R (1.9) All the spectra considered so far - and only these tt - are to be found in the litera- (cf. L. M. LIHTENBAUM (JIiix2), E. V. VAHOVSKIJ (Baxl)). ture; we shall return to this point in the next section. J. J. SEIDEL (LiSe) defines a modified adjacency matrix 8 = (SiT.) for (schlicht) We observe that all of the spectra dealt with up to this point may be derived graphs in the following way: from systems of linear equations the coefficients of which are connected with local structural properties of the graph in question. But the idea of obtaining systems of if i and k are adjacent numerical invariants by exploiting the solvability conditions for a system of equations SiT. =r 1-1 1 (i k), I if i and k are non-adjacent connected with the graph and depending on certain parameters is not at all restricted (1.10) to the use of linear equations; for example, a most natural way of extending the Sii = O. method consists in the transition to a system of quadratic equations of the form ÂX; = E XiXT. (i E f!) (1.18) Obviously, j.i k.; 8=J-I-2A, (1.11) taking the multiplicities of the adjacencies into account by summing over all pairs J denoting a square matrix all of whose entries are equal to 1. t of different edges (arcs) which have i as a starting vertex. In terms of the adjacency The system of linear equations, the characteristic polynomial, and the spectrum t The name matrix of admittance is taken from the theory of electrical networks: any multi- graph G may be considered as corresponding to a specia,l electrical network all branches of which have admittance (= conductivity) 1. t Obviously, if S is the Seidel matrix of the graph G and S is the Seidel matrix of the graph G tt In addition, of course, mention should be made of the "distance polynomial" and corre- complementary to G, then simply S = -So sponding spectrum; see Section 9.2.

i,\ 28 1. Basic properties of the spectrum of a graph 1.3. Some remarks concerning current spectra 29 matrix A = (aik), (1.18) can be expressed in the following form: Then PuCA) = IU - AI = FG(À, 0), (l.21) ÀXi2 = L" aijaikXjXk (aij) + L2 Xj (i = 1,2, ..., n). (1.9) 1 1 i~j~k~" j=I 2 ( 1.22) QG(À) = IDI-IJ.D - AIIDI = - FG(O, À), (The right-hand side of (1.18) and (1.19) is nothing other than the elementary sym- metric function of the second order of all Xk corresponding to the (front) neighbours RG(À) = IU - D - AI = FG(J" -1), ( 1.23) of i, taking into account the multiplicities of the adjacencies.) (1.24) The set of all values of À for which (1.18) and (1.19) have solutions consists of all 0G(À) = IÀl- D + AI = (-1)" FG(-À, 1). zeros of the resultant RG(À) of the system (1.18): so the polynomial RG(J,) and the As for the Seidel spectrum, we can only state system of the roots of the equation RG(À) = 0 (condition of compatibility) can be considered as a characteristic polynomial "of quadratic origin" and the corresponding "quadratic" spectrum, respectively. SG(À) = IU - 81 = (-1)". 2"FG*( À+1 -~' 0 ) Instead of a system of quadratic equations a system of cubic (biquadratic, ...) equations could be taken into consideration, and if G is not regular we may connect = (-1)" . 2" P G* ( _ À ~ 1); (1.25) a system of homogeneous equations depending on more than one parameter (one parameter for each degree, see. next section) with the graph G thus obtaining a here G* stands for a "generalized graph" with weighted adjacencies having the characteristic polynomial depending on several variables. We may even leave the 1 field of algebra and connect with G a system of suitably chosen functional equations "adjacency matrix" A - - J. 2 (boundary value problem, system of integral equations, .. .),t thus obtaining also spectra with infinitely many eigenvalues: the possibilities of connecting "spectra" Remark. FG(À, ll) may be considered as a characteristic polynomial depending w~th graphs are many and varied. on two variables. But (1.20) is, of course, not the only possible way of introducing a It would be very desirable to learn something about the correlations between characteristic polynomial depending on several variables: If, for example, G is non- these different kinds of spectra and especially about the particular role which the regular with s different (out-)degrees Vi' V2, ..., Vs' we make a parameter )'a correspond "linear" spectra play among them: Perhaps it may be possible to specify some to every vertex i with (out-)degree di = Va (0" = 1,2, ..., s). Let À(i) denote the finite system of suitable spectra of a graph G, which, taken as a whole, completely parameter belonging to the vertex i (i.e., ),(i) = Àa with 0" satisfying di = Va) and put characterize G. Interesting as these problems are, they seem to be difficult ones, tt and, since there are at present scarcely any known results worth mentioning, we shall confine our- selves in this book to investigations concerning linear spectra, as described above. A = À (2). ' (1.26) (À(i)o ).(,,) 0) 1.3. Some remarks concernig current spectra then we may generalize PG(À) = IU - AI to PW'I, À2. ..., )'8) = IA - AI. (1.27) All spectra commonly used have been listed in the preceding section; it may be worth mentioning that all of them can be derived from a common source (the Seidel By the specialization spectrum playing a somewhat exceptional role): Put Àa = À + llva (a = 1,2, ...,8), (1.28) FG(À,ll) = IÀl + llD - AI. ( 1.20) i.e., J.(i) = À + lldi (i = 1,2, ..., n),

t A first step in this direction can be found in (PeS 1) (note that formulas (3) and (6) of (PeS 1) from (1.27) the polynomial FG(À, ll) is retrieved: are incorrect, they should be replaced by the above formula (1.19)). See also (Sac 15). tt Experimenting techniques applied to resultants of systems of non-linear algebraic equations F G(À, ll) = P~(À + llVI' À + /tV2, ..., À + llvs) ( 1.29) wil hopelessly fail as the orders of the resulting polynomials are in general beyond any reason- able size - even in simple cases. which is also valid in the regular case. 1.4. The coeffcients of P G(Ã) 31 30 1. Basic properties of the spectrum of a graph

It would certainly be an interesting though possibly diffcult task to investigate components of æO are positive, it follows from Theorem 0.3 that l' is the maximal the significance of these generalized characteristic polynomials, but we shall not eigenvalue contained in the P-spectrum of G. pursue such questions in this book. (See also Section 4.5.) It is worth mentioning that there is stil another important class of multigraphs for which the spectra Spp(G) and SPQ(G) are equivalent, namely, the class of semi- We return to formulas (1.21)-(1.25) and assume G to be a (multi-)graph which is regular of a certain degree 1": we shall show that in this case the four spectra regular multigraphs of positive degrees. (Recall: a multigraph G is called semi- Spp, SPQ,-i SPR, Spc are equivalent, i.e., contain the same amount of information 1"egula1' of degrees 1"i, 1"2' if it is bipartite having a representation G = (g(i, g(2; OZt) with ¡g(il = ni, 19(21 = n2, ni + n2 = n, where each vertex x E eli has valency ri about the structure of G, and that "almost the same" is also true for Sps' This is quite obvious in the first four cases: Since D = 1", we have and each vertex x E g( 2 has valency 1'2') In this case, a straightforward calculation shows that the vector æO = (Vdi, Vd2, ..., vtl,T (with d¿ = 1"i or 1'2) is an eigenvector U + tlD = (À + 1'tl) 1 of the adjacency matrix of G belonging to the eigenvalue V1"11"2, and since all com- and consequently ponents of æO are positive, it follows again from Theorem 0.3 that V1"i1'2 is the maximal eigenvalue. Recall that the maximal eigenvalue is called the index of G denoted FG(À, tl) -- FG(À + 1'tl, 0) = PG(À + 1"tl). ( 1.30) by /2. According to (1.5)" (Section 1.2), So, according to (1.22)-(1.24), Qi;(J,) = IJ. -A*! = J. - '"ri- A = -IÀ/21 on~ - AI =~on, - PG(q:1). 1 I 1 \ 1 1 QG(À) = -PG(1"), (1.31) 1"n So we have proved

RG(À) = PG(À - 1"), ( 1.32) Theorem 1.1 (F. RUNGE (Rung)): Let G be a multigraph either 1"egula1" of positive degree 1" or semi1"egula1" of positive deg1"ees 1"i, 1"2' and let /2 be the index of G. Then /2 = l'

GG(À) = (-l)nPG(-À + 1"), (1.33) 01" /2 = ~, 1"espectively, and in either case and from 1 QG(J,) = - PG(/2À). Spp(G) = (À1, À2, ..., Àn)H en we deduce Note that a connected multig1'aph G is 1"egula1" 01" semi1"egular of positive deg1"ee(s) if and only if the line graph of G is 1"egula1". S G (À1 À2 ÀnJ (1.31') PQ( )= -,-,...,-,1" 1" 1"

SPR(G) = (:11 + 1",:12 + 1", ..., J'n + 1"), (1.32') 1.4. The coefficients of P GO. )

SpdG) = (1" - Àn, 1" - Àn-1, ...,1" - J,d. (1.33') In the next three sections we shall be concerned with relating the coefficients of In the case of the Seidel spectrum, by due computation making use of the eigen- Pi;(À), GG(:1), and QG(:1), respectively, to structural properties of the graph G. vectors, we obtain Let G be an arbitrary iiulti-(di-)graph and

PG(J,) = J,n + a1:1n-1 + ... + an SG(J,) = (-l)n. 2n ÀÀ+1+21" + 1 + 21" - n P (_G À +2' 1). (1.34) , its characteristic polynomiaL. It has been observed by several authors-i that the the eigenvalues with respect to S are -2:1n+2-; - 1 (i = 2, 3, ..., n) and, in addition, values of the coefficients ai can easily be computed if the set of all directed cycles n - 21" - 1. (See also Section 6.5, Lemma 6.6.) of G (considered as a digraph) is known. The converse problem of deducing structural If G is regular of degree 1", then, as can easily be checked, æO = (1, 1, ..., 1)T is an properties of G (for example, concerning the cycles contained in G) from the values eigenvector of its adjacency matrix A belonging to the eigenvalue 1', and since all of the a¡ is much more difficult; we shall return to this problem in Section 3.1.

-i Here l' ? 0 is assumed. H Note that Ã1 = r (see Section 0.3). -i See the remark on the history of the "coefficients theorem" (p.36). 32 1. Basic properties of the spectrum of a graph 1.4. The coefficients of P a(),) 33

The following theorem is sometimes called the "coefficients theorem for digraphs". Then

( -1)i ai = L, b( U) . ( 1.37) Theorem 1.2 (M. M:ILIC (Mili), H. SACHS (Sac2), (Sac 3), L. SPIALTER (Spia )t): Let UEo¿t,

PG(),) = I.U - AI = Â,n + alÂ,n-i + ... + an Proof of Theorern 1.2. Let us first consider the absolute term be the characteristic polynomial of an a1'bitrary (directed) multigraph G. Then an = PG(O) = (-I)n IAI = (-I)n l(tikl' According to the Leibniz definition of the determinant, ai = ~ (_l)p(L) (i = 1,2, ..., n) (1.35) LEse, an = "'(-I)n+I(P)al,a2'£. 11 12 nin...a. (1.38) p where .!i is the set of all linear dÍ1ected subgraphs L of G with exactly i vertices; p(L) denotes the number of components of L (i.e., the number of cycles of which L is com- with summation taken over all permutations posed). This statement may be given the following form: p=ii i2 ... in' ' The coefficient ai depends only on the set of all lÙiear directed 8ubgraphs L of G (1 2 .. . n) having exactly i vertices, the contribution of L to ai being + 1 if L contains an even, I(P) denotes, as usual, the parity of P. For the sake of simplicity, let us first and -1 ,if L contains an odd, number of cycles. assume that there are no multiple arcs so that aik = 0 or 1 for all i, k. A term If G is an undirected multigraph, we may stil consider G as a multi-digraph Gf Sp = (_I)n+I(P) (tlii a2i,'" anin (see Section 0.1, p. 12); all that is necessary to observe is that to every edge of G which is not a loop there corresponds a cycle of length 2 in Gf, and to every circuit of the sum (1.38) is different from zero if and only if all of the arcs (1, iil, (2, i2),..., of G there corresponds a pair of cycles in Gf, oriented in opposite directions. Theorem (n, in) are contained in G. P may be represented as a product 1.2 may now be easily reformulated for multigraphs as follows: P = (1i1 ...) (...) ... (...) Theorem 1.3 (H. SACHS (Sac2), (Sac3), L. SPIATER (Spia)*): Let of disjoint cycles. t P G(Â,) = IU - AI = Â,n + alÂ,n-i + ... + an Evidently, if S p =! 0, then to each cycle of P there corresponds a cycle in G: thus to be the characteristic polynomial of an arbitrary undirected multigraph G. P, there corresponds a direct sum of (non-intersecting) cycles containing all vertices Ca.l an "elementary figure" of G, i.e., a linear directed subgraph L E .!n- Conversely: To each lineal' directed a) the graph K2, or subgraph L E .!n there corresponds a permutation P and a term Sp = ::1, the b) every graph Cq (q ~ 1) (loops being included with q = 1), sign depending only on the number e(L) of even cycles (i.e., cycles of even length) call a "basic figure" U every graph all of whose components are elementary figures; among all cycles of L: let p(U), c(U) be the number of components and the number of circuits contained in U, Sp = (_I)n+e(£). respectively, (ind let óli denote the set of all basic figures contained in G having exactly Obviously, i vertices. Then n + e(L) _ p(L) (mod 2) ai = ~ (-l)P(U) .2c(U) (i = 1,2, ..., n). (1.36) hence UEo¿l, an = ~ Sp = ~ (_I)p(L). (1.39) This theorem may be given the following form: P LEse n Define the "contribution" b of an elementary figure E by Now, (1.39) remains valid even if aik ? 1 is allowed: Consider the set of all distinct linear directed subgraphs L E .! n connecting the n b(K2) = -1, b(Cq) = (-l)q+l. 2 vertices of G in exactly the way prescribed by the cycles of a fixed permutation and of a basic figure U by P = (lil ...) (...)... (...). It is clear that this set can be obtained by arbitrarily choosing for each k an arc from vertex k to vertex ik; and doing so in every possible b(U) = IT b(E). EcU t Note that I(P) == e(P) (mod 2), where e(P) is the number of even cycles among all cycles t See the remark on the histOry of the "coefficients theorem" (p. 36). of the cycle representation of P given above.

3 Cvetkovic/Doob/Sachs 1

34 1. Basic properties of the spectrum of a graph 1.4. The coeffcients of PGP.) 35 manner; and since for fixed k there are exactly aki. possible choices, the total number be the perm-polynomial belonging to an arbitrary iindÙ'ected multigraph G with ad- of subgraphs so obtained equals ali, a2i, ... anin' Thus the total contribution of jacency matrix A. Then all of these subgraphs to the sum ~ (- l)p(L) equals (- 1 )n+I(P) aii,a2i, ... anin' a,¡ = ~ 2c(U) (1.36*) LE,f n (i = 1,2, ..., n). Summation with respect to all permutations P confirms the validity of (1.39) in UEÚl(, the general case. Theorems 1.2 and 1.2* may be extended. to digraphs with weighted adjacencies In order to complete the proof of (1.35) suppose 1 ~ i ~ n (i fixed). It is well immediately: known that (-1)i ai equals the sum of all principal minors (subdeterminants) of Suppose that adjacency k . i has (arbitrary) weight aik't and let A = (aik) be the order i 'of A. Note that there is a (l,l)-correspondence between the set of corresponding generalized adjacency matrix. Then Theorems 1.2 and 1.2* stil these minors and the set of all induced subgraphs of G having exactly i vertices. By hold with applying the result obtained above to each of the (~) minors, and summing, the ai = ~ (_l)p(i) II (L) (i = 1,2, ..., ii) (1.35)' LE,f, validity of (1.35) is established. i and Remark. If, instead of the determinant, the permanent of A, a¡ = ~ II (L) (i = 1,2, ..., n) (1.35*)' LE,f , Per A = '".t ai' ii a2' i2 ... anin, ' p instead of (1.35), (1.35*), respectively, II (L) denoting the product of the weights of all arcs belonging to L. is considered, we obtain by means of analogous deductions the simple formulas If G is an undirected graph with weighted adjacencies and U is a basic figure per A = number of directed linear factors-i of G, (1.40) contained in G, let and in the case of an undirected multigraph: II (U) = II (w(u))~(U;ul, UEE(U) per A = ~ 2c(U) . (1.41 ) UEq¡ n where E(U) is the set of edges of U, w(u) is the weight, of the edge u, and

Call perm-polynomial of an arbitrary square matrix A of order n the polynomial C(u;U 1 1) if =u is'. contained in some circuit of U, per (U + A) = J," + a~Ji"-l +... + a:. 2 otherwise. ' Since U contains exactly 2c(U) linear directed subgraphs L all having the same weight The analogues of Theorems 1.2 and 1.3 are then: II (L) = II (U), (1.35)' takes the simple form Theorem 1.2*: Let Ui = ~ (-1)P(U) 2c(U) II (U). (1.35)" UEúj(, P"((Ji) = per (U + A) = Ji" + aiJ,"-l + ... + a~ With i = n, we obtain from (1.35)' a simple formula for the calculation of the be the perm-polynomial belonging to an arbitrary (directed) multigraph G with ad- determinant of an arbitrary square matrix A considered as a generalized adjacency jacency matrix A. Then matrix of a digraph G: ¡AI = (-1)" ~ (_l)p(L) II (L) (1.42) at = number of linear directed subgraphs of G LE,f n

containing exactly i vertices (i = 1,2, ..., n). ( 1.35*) (note that .! n is the set of all dirécted linear factors L of G). If, in particular, A, is the adjacency matrix of a multi-digraph or a multigraph, Theorem 1.3*. Let (1.42) reduces to

P"G(J,) = per (U + A) = Jin + aiJi"-l + ... + a: ¡AI = (-1)" ~ (_1)p(L) ( 1.42)' LE,f n t A directed linea'r factor of a multi-(di-)graph G is a linear directed subgraph containing all -i We may assume that for every pair i, k there is exactly one arc from i to k, and that aik is vertices of G. the weight of this arc (possibly equal to zero).

3* 1

1.5. The coefficients of OG(Å.) 37 36 1. Basic properties of the spectrum of a graph or concerning the coefficients of a generalized characteristic polynomial ¡AI = (-1)" L (-1)P(U) 2c(uL, (1.42)" Pp.) = dl.(A - Ål), UEo¿tn respeetively. dl. being a matrix function generalizing determinant as well as permanent given by (1.42) may be taken as an intuitive form of the Leibniz definition of the determinant. A theory of determinants based on this .observation was outlined by D. M. CVETKO- dl.(A) = L X(P) aliia2i2 ... a"in VIó (Cve 15). p with summation over all permutations P = ;. .; here x(P) denotes some Remai'k (concerning the history of the coefficients theorem). In order to show (12...n)i¡ i2 ... i" that this approach is not only of purely theoretical interest, it should be noted that character defined on the symmetric group Y" of all permutations P considered. there are two other fields in which determinants have been connected with graphs: elec- tronics-cybernetics (signal flow graph theory) and chemistry (quantum chemistry, Some simple consequences of Theorems 1.2 and 1.3 simple molecular orbital theory). Apparently (1.42) was given for the first time by C. L. COATES (Coat) (1959) in Proposition 1.1: The numher of linear subgraphs with exactly q edges contained in an connection with flow graph considerations t; (1.42) is therefore sometimes called undirected forest H is equal to (-l)q a2q. An undirected linear factor exists if and only Coates' formula. A simple proof is given by C. A. DESOER (Deso) (1960). F. HARARY if aii =j O. In this Gase, n is even, and, as there evidently cannot be more than one linear n (Har2) (1962) considers the case when A is the adjacency matrix of a digraph or of a graph. But before COATES other authors came close to formula (1.42) factor, an = (-1) 2 . (see D. KÖNIG (Kön1) (1916), (Kön2) (1936); see also T. MuIR (Mui2), footnote on Proposition 1.2: The numbei' of directed linear factors contained in a multi-digraph 0 p. 260 concerning Cauchy's rule for determining the sign of a summand in the is not srnallei' than lalil. expansion of a determinant). For some small values of i, the coefficients ai of the characteristic polynomial of an The general problem undirected graph 0 were already determined by C. A. COULSON (Cou2) (1949) and "Let the characteristic polynomial PG(À) of some multi-(di-)graph 0 be 1. SAMUEL (Sam1) (1949) (see also (Sam2)) in the context of molecular orbital given, what information about the cycles (or circuits) contained in 0 can theory, and, independently, by L. COLLATZ and U. SINOGOWITZ in their fundamental be retrieved from the coefficients ai ~" paper (CoSi1) (1957)tt on graph spectra. COULSON (Cou2), however, does not use wil be treated in Chapter 3, Sections 3.1-3.3. the concept of "basic figures" but expresses the coefficients by means of the numbers of all possible subgraphs of 0 with the given number of vertices. In this connection, E. HEILBRONNER'S papers (Heil) (1953), (Hei2) (1954) should also be mentioned; 1.5. The coefficients of CG().) he showed how, in the case of special graphs arising in the molecular orbital theory, the characteristic polynomial can easily be obtained by some intuitive "graphical" Next we shall express the coefficients Ci of the polynomial recurrence procedures. It seems that the coefficients theorem in full generality was first published by GG(J.) = IÅl- 01 = À" + CiÀ"-l + ... + c" (1.43) H. SACHS (Sac3) (1964) (see also (Sac2) (1963)) and almost at the same time by L. SPIALTER (Spia) (1964) (in a terminology appropriate for chemical applications) in terms of the "tree structure" of 0, where 0 is any multigraph (recall that and M. MrLIÓ (Mili) (1964) (in terms of flow graph theory). Later it has been re- 0= D - A = (oiidi - aii) is the matrix of admittance of 0; see Section 1.2). discovered several times: J. PONSTEIN (Pons) (1966), J. TURNER (Turn2) (1968), Let M be any square matrix with rows 1'1, r2, ..., 1'". and columns Ci, C2, ..., Cn, A. BECE (Beiie) (1968), A. MOWSHOWITZ (Mow5) (1972), H. HOSOYA (Hos2) (1972), let.A = 11,2, ..., nl and" = 11'1' 1"2' ..., 1'ql c .A; let M f denote the square matrix F. H. CLARKE (Clar) (1972); for trees it has also been given by L. LovÂsz and J. obtained from M by simultaneously cancelling rows rii' ri2' ..., riq and columns PELIKÂN (LoPe) (1973). TURNER'S paper contains a somewhat more general theorem Cii' ci2' ..., cii For the sake of convenience write ltli instead of M¡i)' etc.; as usual, the determinant of the empty matrix (case" = .A) is assumed to be 1. t With regard to signal flow graph theory, see the fundamental papers of C. E. SHANNON (Shan) (1942) (which remained unnoticed for several of years) and S. J. MASON (Mas1) If 0 is any multigraph with n vertices 1,2,..., n, and if ß =j ø, let 0" denote (1953), (Mas2) (1956); for applications see C. S. LORENS (Lore) (1964). For proofs see also the multigraph obtained from 0 by identifying (amalgamating) the vertices 1'1' 1'2' ..., R. B. ASH (Ash) (1959) and A. NATHAN (Nath) (1961). A detailed treatment may be found in l' q' thereby replacing the set l1'ii 1'2' ..., 1'ql by a single hew vertex i (by this process the book of W.-K. CHEN (Chen) (1971). tt Note that this paper had already been prepared during World War II, see (CoSi2). multiple edges and loops may be created); evidently, 01 = O2 = ... = 0" = O. 1

1.5. The coefficients of GG(À) 39 38 1. Basic properties of the spectrum of a graph

A theorem for multi-digraphs with weighted adjacencies generalizing Theorem The following well-known important theorem connects the number of spanning trees of a multigraph with its matrix of admittance. 1.4 was proved by M. FIEDLER and J. SEDLÁCEK (FiSe). n Remark. For i = n - 1, (1.47) yields Cn-I = (_l)n-l L t(G¡) = (_l)n-l nt(G). :ilatrix-Tree-Theoremi": Let G be a multigraph with vertices 1,2, ..., n and let t(G) j=1 denote the number of spanning trees contained in G. H Then Hence t(G) = ICil, (1.44 ) where C = D -A is the matrix of admittance of G and j E (1, 2, ..., nJ. (i) t( G) = .. (n -1 )n-l Cn-l'

Corollary: Let .I c .A, .I =l ø . Then Let l-1' l-2, ..., l-n (in some order) be the eigenvalues of C. Sincen-l Cn = IA - DI = 0, it follows that 0 E SPc( G); let l-n = O. Then ( _l)n-l Cn-l = n fli, and from (i) t(G .I) = IC .II. (1.45) i=1 Proof of the Corollary. If C' denotes the matrix of admittance of G .I' then C; = C .I' 1 n-l t(G) = - n l-i ar¡d according to the Matrix-Tree-Theorem, t(G f) = IC;I = IC ß:I' (ii) n i=1 In the sequel the convention t(Gø) = 0 wil be adopted so that (1.45) holds for every .I c .A (note that ICøl = ~e¡ = 0). is obtained. Now we are in a position to calculate the coefficients Ci of 0o(À) = IÀl - e¡. If G is connected, t(G) ? 0, i.e., l-i =l 0 for i = 1,2, ..., n - 1. Thus we have proved Since (-I)i Ci is equal to the sum of all principal minors of order i of C, cn_k=(-l)n-k L IC.l1 (k=O,l,...,n), (1.46) Proposition 1.3: Let G be a connected multigraph. Then 1 fI.li~k c,A t(G) = - II l-, i n where, according to the corollary of the Matrix-Tree-Theorem, IC.l1 equals t(G .I), Thus we have proved where l- runs through all non-zero eigenvalues of C = D - A.

Theorem 1.4 (A. K. KEL'lVIANS (KeJI3)): Let In terms of the polynomial Oo(À) or the Kel'mans polynomial B~(G) (see (1.17), Section 1.2), this result can also be expressed in the following form: 0o(À) = IÀl - e¡ = co).n + ciJ,n-i + ... + Cn (co = 1), , (_l)n-l 1 1 n where G is an arbitrary multigraph and C = D - A is its matrix of admittance. Then (iii) t(G) = OG(O)n = n- Bo(G). Ci = (_l)i L t(G.I) (i = 0, 1, ..., n). (1.47) If G is regular of degree 1', formulas (1.33) and (1.33') apply and we deduce from (i), Le",y, i;)'-i=n-i (ii), and (iii) (recall that À1 = 1') Let the forest F have k components Ti with ni vertices (i = 1,2, ..., k) and put Proposition 1.4 (H. HUTSCHENREUTHER (Huts)): For any regular multigraph G y(F) = nin2 ... nko According to (KeCh) (see formula (2.14) on p. 203), c¡ can be given of degree 1', the following form: t(G) = -1 II (1'n - Ài) 1 = - P~(r), Ci = (_1)i 1. y(F) (i = 0, 1, ..., n - 1), Cn=O, (1.47)' n i=2 n FE,% n-' where the Ài are the ordinary eig€nvalues of G.

where ?k is the set of all spanning forests of G with exactly k components. By adding an appropriate number of (simply counted) loops, any multigraph G of maximal valency r can be made a regular multigraph G1 of degree 1'. Since this i" This theorem was proved in a paper by R. L. BROOKS, C. A. B. SMITH, A. H. STONE, and W. T. TUTTE (BrSST) (1940), and independently by H. M. TRENT (Tren) (1954), and others; process has no influence on the number of spanning trees, Proposition 1.4 can be an elementary proof was given by H. HUTSCHENREUTHER (Huts) (1967). Some authors hold applied to an arbitrary multigraph G, provided the À¡are taken to be the eigenvalues that it is already implicitly contained in G. KIRCHHOFF'S classic paper (Kirc) (1847). (For not of G but of G'. This observation, due to D. A. WALLER ((Wall), (Wa12), (Wa13); more details consult (Mo02) (Chapter 5).) see also (Ma12)), is equivalent with Proposition 1.3. H t(G) is sometimes called the complexity of G. - A simple determinant formula for the complexity of a bipartite graph is due to F. RUNGE; see Section t.9, no. 12. (See also Section 1.9, nos. 10, 11.) 1 1.7. Cyclic structure and tree structure 41 40 1. Basic properties of the spectrum of a graph

1.6. The coefficients of QGo.) Theorem 1.5 has also been extended to graphs and digraphs with weighted ad- jaeencies by F. RUNGE (Rung). By a procedure very similar to the method used in the proof of the preceding theo- RernaTk. In order to obtain a coefficients theorem for QG(À) based on the cyclic rem, the eoefficients of QG(À) can be determined. (Recall: QG(À) = ~ IÀD - AI structure of G, recall that QG(À) = IU - A*I with A* = ( a:ik ) (see (1.5)", Section = qoÀn + qiÀn-1 +... + qn (qo = 1); see Seetion 1.2.) IDI Let G be an arbitrary multigraph without isolated vertices. Consider QG(À) as a 1.2). Vdjdk polynomial in .1 - 1: Now formula (1.35)" (Section 1.4) when applied to A* yields immediately

QG(J,) = !U - D-1AI = 1(.1 - 1) I + D-1(D - A)I qi = ~ (- 1 )p(U) 2c(U) II (U) UEo/I, = 1(.1 - 1) I + D-1C¡ = qo(À - l)n + qi(À - 1)n-1 + ... + qn, with where qi equals the sum of all principal minors of order i of D-1C. Accordingly, II(U)=(j.k)E,g(U)( 1II )Ç((i.k);U) V-= djdk =-,II 1 dh qn-k = ~ I(D-1C) fl (k = 0,1, ..., n) hE"f'() fIfl~k c,A where g(U), "f(U) denote the sets of edges and of vertices of U, respectively. Thus with we have proved . IC"I_ t(G f) 1(D-1C)fl =J(D-1)fC fl = ID fl - Theorem 1.5a: Under the assumptions of Theorem 1.5, II di' IE,A- f 2c(U) qi = ~ (_l)p(U)_. (1.48 a) the last equation following from the Corollary to the Matrix-Tree-Theorem (Section UEilll, II dh hEf/(U) 1.5) (if f = %, then II di = 1 is assumed). Thus IE,A - f _ t(G ,,) 1.7. A formula connecting the cyclic structue and the tree structure q,,-k = ~ - (k = 0,1, ..., n), of a reguar or semieguar multigraph fc,AIfl=k IE,A-II di f There are two strong connections between structural graph theory and linear algebra: and since g.-k = i: (j) (-l)i-q"_j' we obtain with k = n - i: j=k k The first one consists of the fact that the most important general invariant of linear algebra, the determinant, may be given a combinatorial form (viz., the form it has qi = (_l)n-i ~ J. (-l)j~. ~ (i = 0,1, ..., n). in its "Leibniz definition") that has an interpretation in terms of the cyclic structure j=,.-i,. ( n .- i) f;:t(G II )d of a (di-)graph (with weighted adjacencies), and the second one is the validity of the l"l~j lEA/"- J Matrix-Tree-Theorem (see Section 1.5) which, in a very simple way, connects the So we have proved tree structure of a' graph with determinants formed from its matrix of admittance. Both Theorem 1.5 (F. RUNGE (Rung)): Let of these connections are taken advantage of by spectral theory: the coefficients theorems for PG(À) (Theorems 1.2, 1.3) are based on the first one, and for GG(À) (Theorem 1.4) 1 and QG(J,) (Theorem 1.5) on the second one. QG(À) = - IÀD - AI = qo),n + qi),n-1 + ... + q,. (qo = 1ì, IDI Of particular interest are those graphs G which have the property that their polynomials PG(),) and GG(J,) or QG(À) can be transformed one into another: in this where G is an aTbitraTY rnultigraph without isolated vertices. Then case, the coefficients can be expressed both in terms of the cyclic structure and in terms of the tree structure of G, thus linking the basic structural elements, cycle q,,-- - -1( ).: n-i . '7( )-1 _ (iJ '7= 0,1,f . ..., n), (or circuit) and tree, one to another. j=,.-i, ,. n( -j) i fcJV, t( IIG di) (1.48) According to Theorem 1.1 (Section 1.3), Ifi~J IE,A- f 1 QG(J,) = -; Pa(e),) ( 1.49) where the conventions t(Gø) = 0 and II di = 1 are adopted. e IEØ 43 42 1. Basic properties of the spectrum of a graph 1 1.8. On the number of walks ! for any multigraph G which is regular or semiregular of positive degree(s) l' or ri, 1'2, and (1.36) and (1.47) now yield the following system of equations equivalent with respectively, and has index (= maximal P-eigenvalue) (2, where (2 = l' or e = Vrir2, (1.50) : respeetively. From (1.49) we deduce L t(Gf)~~,( f .)ri+i-n'L. (_1)p(U)2c(U) (i=O,l,...,n), (1.50') eiqi = ai (i = 0,1, ..., n), i;;î=n-i"tc.f l=n-i n - i UEOJI,,-j and applying Theorems 1.3 and 1.5, we obtain the following theorems. where for f = n the last sum is taken to be 1. With i = n - 1 we obtain from (1.50') a new formula for the number of spanning Theorem 1.6 (F. RUNGE (Rung)): Let G be a regular multigraph of positive degree l' trees contained in a regular multigraph, namely with n vertices 1, 2, ..., n. Then 1 n t(G) = - ~ f' ri-1 L (_l)p(U) 2c(U) ( 1.54) UEOJI,L (-l)P(U) 2c(U) j~n-i = t ( f .)n (_lyi+i-nri+i-n- i ,jc.f L t(G f) (i=O,l,...,n), n i=i UEÓll n-j ifl=j (1.50) (see also Proposition 1.4). where for i = 0 the left-hand sum is taken to be 1. Remark 2. A general formula connecting cyclic strudure and tree structure of any Theorem 1.6 a (F. RUNGE (Rung)): Let G = (!!, qy; 0lt) be a semiregular multigraph, multigraph is, of course, contained in Theorems 1.5 and 1.5a (Section 1.6): From where all vertices x E !! = iI, 2, ..., ni L have valency ri :; 0 and all vertices y E qy (1.48) and (1.48a), after multiplication by IT d¡ we obtain = ini + 1, ni + 2, ..., n1 + 1i = n; have valency 1'2:; O. l'hen Theorem 1.7: Let G = (!!, 4't) be any multigmph withoid isolated vertices, where for odd i E JV, !! = JV = r1, 2, ..., n). Then

, f,( f .)(-l)j ~ r~ir~'t(Gf)=O, L (-l)P(U) 2c(U) IT d/¡ ~ £ ,( f .) (_l)n-i+j 2: t(G f) IT di i~n-i n - i fc.f (1.51) UEiíll, hE.f-i/'(U) i=n-i n - i fc.f IEf ifl=j ifl=i (lnd for even i E JV, (i = 1,2, ..., n), where "Y(U) is the set of vertices of the basic ligiire U and where the i . -n conventions t(Gø) = 0 (in(Z IT di - 1 are adopted. n ( 7' ) i.+ii-ni -;+12- 2 t(G tE), ~ (-l)p(U) 2C(U) = ~ (_l)i+j-n '" 1'2 . 1'- d IEØ UEOJI, i=n-i n - i ,jc.f 2 ~ .:. ,/ i By specialization Theorems 1.6 and 1.6a are obtained from Theorem 1.7, but in ifl=j (1.52) the general case the significance of Theorem 1.7 is constrained by the fact that the where in the last sum of (1.51) and (1.52) 1i = l!! nfl, f2 = iqy n fi (i1 + f2 = f). terms depending on the valencies d/¡ or ell cannot be eliminated. Rema~k 1. For regular multigraphs of positive degree 1', we may use the relation

Go(J,) = (-l)n PG(-), + 1') ( 1.33) 1.8. On the number of wals

(Section 1.3) instead of (1.49), equate corresponding coefficients and apply Theorems In this section, "spectrum" always means "P-spectrum". 1.3 and 1.4 (instead of 1.5). The relation connecting the coefficients ai of PG().) and Let A be the adjacency matrix of a multi-digraph G with vertices 1, 2, ..., n. If, in Cj of GO(Å) is addition to the spectrum of G, the eigenvectors of it. are known, then, of course, more , ai = (- l),i I: ,( f .) ,/i+i-n Cn-j (i = 0, 1, ..., n) (1.53) statements concerning the structure of G can be made than without this knowledge. I i~n-i n - i , Moreover, a multi-digraph G with a symmetric adjacency matrix - in partiçular, a multigraph - is completely determined by its eigenvalues and eigenvectors. For, with ao = Co = 1, and with (1.36) (Theorem 1.3) and (1.47) (Theorem 1.4) we arrive if Vi, Vi, . ", Vn is a complete system of mutually orthogonal normalized eigenvectors again at Theorem 1.6. of A belonging to the spectrum (I'i, )'2' ..., )'n), let 17 = (vi, V2, . ", vn) = (Vij) and By inversion of (1.53) we obtain A = (åijÅi):A then, =as is well 17 known, ITAVT. is orHiogonal(i.e., y-i (1.55) = VT) and Ci_ - (-1)in (fL, .).i+j-n 1 an-j (i - '_0,1, ..., n), ( 1.53') i=n-i n - i Since G is determined by A, we have proved 44 1. Basic properties of the spectrum of a graph i 1.8. On the number of walks 45 i Theorem 1.8: A multigraph is completely determined by its eigenvalues and corre- in G (k = 0, 1,2, .. .). Then sponding eigenvectors. So, in principle, any multigraph problem can be treated in terms of spectra and eigenvectors. (For example, an algorithm for determining whether two graphs are H,l') ~ : ii-i/õ ~~;r) - iJ ( 1.59) isomorphic, which is based on Theorem 1.8, has been developed in (Kuhn).) From this point of view, we shall now investigate the problem of the number of walks of . given length in a multi-(di-)graph G~ (Recall: A walk of length k ~ 0 is a sequence Proof. If M is a non-singular square matrix of order n, let Pl1l denote the matrix of arcs U1U2'" 'uk- where the starting vertex of Uj+l coincides with the end vertex of Uj formed by the minors of order n - 1 so that (WIlT = IMI M-1. Let sum M denote (j = 1,2,..., k - 1), repetitions and loops being allowed.) Some more problems concerning eigenvectors wil be considered in Section 3.5. the sum of all elements of M, and let J be a square matrix all entries of which are equal to 1; then, for an arbitrary number x, The starting point of our considerations is the following well-known theorem. sum (M) ( 1.60) 1M + xJI = IMI + x Theorem 1.9: Let A be the adjacency matrix of a multi-digraph G with vertices 1,2, ..., n, let Ak = (air); further, let Nk(i, j) denote the number of walks of length k which can be proved by straightforward calculations. Now, according to Theorem 1.9, starting at vertex i and terminating at vertex j. Then Nlc = sum A Ie and since Nlc(i, j) = aijk) (k = 0, 1,2, ...). (1.56) 00 ~ Aktk = (I - tAt1 = II - tAI-1 (I - tAl (It i ~. (max 2i)-1), Note that for k = 0, (1.56) agrees with the convention No(i, j) = Öij' k=O Now let G denote a multigraph and let V = (Vij) be an orthogonal matrix of eigen- we obtain vectors of A, as described above. Then, according to (1.55), 00 00 ~ sum Aktk = ~ Nlctk = II - tA!-l sum (I - tAl, (k) n~ ,k k=O k=O aij - ~ VivVjvJlv . i.e., v=1 ( 1.57) The number Nlc of all walks of length k in G equals H G(t =) sumII -(I tAl - tAl (1.61) n (n )2 Nlc = tr Nlc(i, j) = ti air = P~l iE Vip 2:. With M = I - tA, x = t, (1.60) yields Thus we have proved 1 ( - sum (I - tAl = - I(t + 1) I + tAl - II - tAl), ( 1.62) Theorem I.IO:t The total number N k of walks of length k in a multigraph G is given by t " Nlc = ~ C,ì.: (~ = 0, 1,2, ...), where A = J - I - A is the adjacency matrix of the complement G of G, and by v=1 ( 1. 58) inserting (1.62) into (1.61), the equation

where Cp = ,~Vip . ("i~l )2 -- i - A HG(t) = ~ (_1)" t - 1 ( 1.63) In the next theorem, the generating function for the numbers Nlc is expressed in t \~I-AI terms of the characteristic polynomials of the graph G and its complement G. Theorem 1.11 (D. M. CVETKovró (Cve8)): Let G be a graph with complement G, and L i t + 1 -\ L 00 is obtained. Clearly, (1.63) implies (1.59), which proves the theorem. let HG(t) = ~ Nktk be the generating function of the numbers Nk of walks of length k k=O Theorem 1.11 has been proved in (Cve8) by another method. P. W. KASTELEYN

t Part of this theorem was proved in another way by D. M. CVETKovrc (Cve9) who also IKas2) gave the expression for the generating function for numbers of walks between proved the theorem in the present form when preparing the manuscript of this book; the two prescribed vertices of a graph. theorem was also partly used in (CvS 1). Another proof was given by F. HARARY and A. J. The generating function HG(t) wil be used in Section 2.2. The numbers of walks SCHWENK (HaS 1). for graphs of some special types wil be determined in Section 7.5. ,. I i 46 1. Basic properties of the spectrum of a graph 1.9. Miscellaneous results and problems 47

Let (,ui, ,u2, ..., ,urnI be the set of distinct eigenvalues of a multigraph G. (1.58) can then be rewritten in the form 1.9. Miscellaneous results and problems 1. Let G be a multi-(di-)gmph with vertex-set 11,2, ..., nj and let Nlc(i, j) denote the number Nlc = Di,uf + D2,u~ +... + Drn,u~ (k = 0, 1,2, ...), (1.64) of walks of length k in G joining i to j. If ivii is the corresponding generating function (i.e., where Di, D2, ..., Drn are non-negative numbers uniquely determined by G; some 00 wii = k=O~ Nlc(i, j) tk)(P. and TV = (wii)'W. thenKASTELEYN TV = (1 - tA)-l. LKas2J) (but not all) of them may be zero. In particular, with k = 0 the equation Ci + C2 +... + Cn = Di + D2 +... + Dm = No = n (1.65) , JI+1 2. Let IR be the set of the greatest eigenvalues of all graphs. Let T - - (the golden mean). For n = 1, 2, ..., let ßn be the positive root of 2 is obtained from (1.58) and (1.64). Pn(x) = Xn+l - (1 + x + x2 + ... + xn-l).

D. M. CVETKOVIC (Cve9) gave the following Let IXn = ß~2 + (3112. Then 2 = IXl ~ IX2 -c ... are all limit points of IR smaller than 7:1/2 + .-i12 = lim IXn- Definition. The main part 01 the spectrum of a multigraph G is the set of all those n-++oo eigenvalues ,uj for which in (1.64) Dj =1 0 holds. (A. J. HOFFMAN LHof13)) 3. If a digraph G has at least one cycle then the index of G is not smaller than 1; otherwise all For a regular multigraph of degree r with n vertices, clearly, Nlc = nrk: hence, for eigenvalues of G are equal to zero. regular multigraphs (and, in fact, only for these) the main part of the spectrum con- (J. SEDLÁCEK LSed 1)) 4. Let G be a digraph with vertices 1, ,.., n. For given vertices i and j (i =F j), a spanning sists of the index only. In this case, ~lc = T which motivates 1 :lc in the gene- sub graph of G in which 1 ° exactly one arc starts and no arc ends in i, ral case to be considered as a certain kindY- of mean value k/-of the valencies, in general depending on k. This gives rise to the following 20 exactly one arc ends and no arc starts in j, 30 all the other vertices have in and out degrees equal to 1, Definition. Let G be a multigraph and d = d(G) = lim Nlc = lim VNlc (it wil is called a connect'ioii /TOm i to j and is denoted by O(i -)- j). For i = j the vertex i is an iso- . k-'oo n k-'ooY- k lated vertex of O(i -)- i) while all the other vertices have property 3°. be shown that the limit exists). Then d is called the dynamic mean 01 the valencies of With a square matrix A = (aii)~ we associate a weighted digraph DA, defined in the following the vertices of G. way. The n vertices of D A are numbered by 1, 2, ..., n and for each ordered pair of vertices i, j there exists an arc in DA leading from j to i and having weight aii' Clearly, Nlc = O(dk) (k -- (0). The product TV = W(L) of the weights of the arcs of a spanning linear sub graph L is called the weight of L. The number of cycles contained in a linear subgraph L is denoted by c(L); Theorem 1.12 (D. M. CVETKOviC (Cve9)): For a multigraph G, the dynamic 'mean !I denotes the set of all spanning linear subgraphs L of DA. d(G) is equal to the index 01 G. The weight TV(O(i -)- j) and the number of cycles c(O(i -)- j) of a connection O(i -)- j) are defined analogously. Theorem 1.12, together with the existence of d, follows immediately from Theorem Then the cofactor A¡i of the element aij is given by 1. 10 and the fact that among the eigenvectors corresponding to the index of G there Aii = (-1)n- ~ (_1)c(C(i--j) W(O(i -)- j)), is a non-negative one. . C(i-'j) An application of this theorem to chemistry is described in (CvG4). where the summation runs through all connections O(i -)- j) from i to j of the digraph DA. Consider further the following system of linear algebraic cquations: We quote without proof ii ~ aiixi = b¡ (i = 1, ..., n). Theorem 1.13 (F. HARARY, A. J. SCHWENK (HaS 1)) : For a multigraph G, the j=i lollowing statements are equivalent: With this system we associate a digraph D having vertices 0, 1, ..., n in which the vertices 10 JI is the main part 01 the spectrum; 1, ..., n induce the digraph D A, corresponding to the matrix A = (a¡i)'i and in which there is an additional arc from vertex 0 to vertex i having weight -bi for every i E 11,2, ..., nj. Then 20 ~I is the minimum set 01 eigenvalues the span 01 whose eigenvectors includes the vector (1, 1, ...,I)T; ~ (_1)(C(o--j)) W(O(O --j)) xi=-C(o--j) (j = l, ..., n), 30 JI is the set 01 those e-genvalues which have an eigenvector not orthogonal to ~ (_l)c(L) W(L) LE!I (1, 1, ..., I)T. where in the upper sum the summation runs through all connections 0(0 -)- j) in D. The proof can be performed by means of Theorem 1.10. (C. L. COATES LCoat)) 48 1. Basic properties of the spectrum of a graph 1.9. Miscellaneous results and problems 49 5. Let G be the digraph corresponding to a square matrix A of order n (see il. 4). Let M¡i be the cofactor of the i,j-element of 1,1 - A. Then Comllary: If G is either regular of positive degree i' or semiregular of positive degrees 1'1' 1'2' and if 12 is the index of G (i.e., 12 = l' or 12 = ¥i'1i-2, respectively), then 71 Mii = L Àn-k L (_l)c(c,,(i-.j)) W(Ck(i -+ j)), 1 0 ~ ,0 k=2 c.(i-'j) m = - 12" ¿" 1\;. 2 v=1 (1.69) where in the second sum the summation runs through all connections Ck(i -)- j) from i to j Problem. Is condition (1.69) sufficient for a graph G to be either regular or semiregular ? which have exactly k vertices. (F. RUNGE (Rung)) (J_ PONSTEIN (Pons)) 9. The determinant of the adjacency matrix A of a multigraph G is given in terms of the tree 6. Some i-emai-ks concerning the Q-specti-m of a multigraph. If G is a multigraph without iso- structure of G by lated vertices having components G1' G2, ..., Gk then, clearly, 71 k ¡AI = (-1)71 L (-l)i L (n di) t(G f)' QG(À) = n QG,(). j=1 fc.liEf i=1 (1.66) Ifl~j (F. RUNGE (Rung)) QG may now be defined for multigraphs G having isolated vertices in the following way: 10. Let G be a multigraph having n vertices with positive valencies ai, ..., dl/ Then the com- i) If P is the "point graph" having exactly one vertex and no edge, set plexity of G is given by Qp(À) = À - 1. (1.67) t(G) = lQ %(1) = n d¡ IÎ (1 _ i,~'), (ii) If G is any multigraph having components G1' G2, ..., Gh set 2m L di v=2 k where in is the number of edges and where the À: are the Q-eigenvalues of G. QG(À).= n QG;(À) i=1 which is consistent with (1.66). (F, RUNGE (Rung), (RuSa); see also (Sac 12)) 11. Let G = (ít, qy; '1t) be a bipartite multigraph without isolated vertices where If G is a multigraph without isolated vertices, then, according to (1.5)' and (1.5)" (Section 1.2) ít = IX1' ..., xml, qy = (xm+l' ..., xm+nl; let V, W be the valency matrices of the sets ít and qy, QG(À) = IH - ÃI = ¡H -A*I, respectively, so that the adjacency matrix A and the valency matrix D of G are of the form where (1.68) 1 1 A = (0 B), Ã = D-1A = (aik),di A* = D Yd¡dk-"2AD -"2 = ( a¡k ). BT 0 Ð = (VOW 0) , respectively (B is an n X m matrix). Put If G is the point graph P, set A * = Ã = (1), consistent with (1.67). The matrix A * is symmetric, Ã is stochastic, so all Q-eigenvalues of G are real, the largest one, V-IB = lU, W-1BT = M, being equal to 1. The Q-spectrum has many properties analogous or very similar to properties of the P-spec- ai~(À) = IH - MMI, ip~(À) = IH - MMI, trum not to be itemized here; a few examples shall be quoted: where I denotes the identity matrix of order in or n, respectively. Then, by a well-known The number of components of G is equal to the multiplicity of the Q-eigenvalue 1 of G. theorem of the theory of matrices, - Let G be 0, multigraph without isolated vei'tices. G is bipartite if and only if QG( -À) = QG(À). - Let G be a connected multigraph, not the point graph. G is bipartite if and only if QG( -1) = O. Ànai~(À) = Àmip~(À).

Let G be a multigraph and let kG denote the multigraph derived from G by replacing every Put edge by exactly k parallel edges. Let A(G) denote the adjacency matrix of G, etc. Clearly, aiG(À) = A(kG) = kA(G), butA *(kG) = A *(G),Ã (kG) = Ã (G). Thus G, though uniquely determined by A, iai~(À)ip~(À) ifif n ~ m.in, is not determined by A* or Ã. Multigraphs G and kG have the same Q-spectrum. This obser- vation is, of course, not meaningful when only graphs are considered. Thus the order of the polynomial aiG(À) is equal to min (m, n), Note that 'PG(I') is invariant under the interchange of the vertex sets ít and qy. The polynomial aiG(I') is connected with 7. Let G be a multigraph without isolated vertices and put Ãl = (ãW) (i = 1,2, ...). Then QG(À) by the formula ãh1 equals the probability of reaching vertex j as the last point in a random walk of length i starting at vertex i. Àmin(m,n)QG(À) = Àmax(m,n)aiG(À2) 8. Let G = (ít, '1t) be a graph with in edges and n non-isolated vertices having Q-spectrum so that essential information contained in QG(À) is already contained in aiG(À). Thus, for bi- partite multigraphs, it may be more convenient to use aiG(À) (or the corresponding ai-spectrum) Spo(G) = (À1, À2, ..., 1'71)0. Then than QG(À) (or the Q-spectrum). For example, in terms of the ai-spectrum the complexity of G is given by v=1i: À; (i,j)E'1t = 2 d¡L dj-- t(G) = ¡VI.i IWI.~2 IT (1 -L Å.) d¡ = 2 n.=2 d¡ n (1 _ Å.), 50 1. Basic properties of the spectrum of a graph r

where I is the number of edges of G, k = min (m, n), and where the X~ are the rp-eigenvalues of G. If the last formula is applied to the complete bipartite graph Km,n' the well-known ,;¡ 2. Operations on Graphs and the Resulting Spectra formula t(Km,n) = mn-1 . nm-1

(see (FiSe)) is immediately obtained. (See also Section 7.6, p. 219.)

(F. RUNGE (Rung), (RuSa), (Sac12J) 12. Let G = (fl, '!; ql) be a bipartite multigraph without isolated vertices. .With the notation of no. 11, the complexity of G is given by t(G) = ¡Wi, IW - BW-1BT)il = ¡VI . I(W - BTV-1B)jl' where .i, j are arbitrary numbers taken from (1,2, ..., ml or (1,2, .,' " nL, respectively. (Recall that Mi denotes the matrix which is derived from the square matrix M by simultaneously deleting the i-th row and the i-th column.) In this chapter we shall describe some procedures for determining the spectra and/or (F. RUNGE (Rung), (RuSa), (Sac12)) characteristic polynomials of (directed or undirected) (multi-)graphs derived from 13. The considerations of no. 11 may be taken as a starting point for developing a spectral some simpler graphs. In the majority of cases we have the following scheme. Let theory for hypergraphs: Any hypergraph H can be represented by its incidence gmph (Levi graphs Gi, ..., Gn (n = 1,2, ...) be given and let their spectra be known. We define graph) G = L(H) which is a bipartite graph without isolated vertices; conversely, every con- an n-ary operation on these graphs, resulting in a graph G. The theorems of this nected bipartite graph G with more than one vertex uniquely determines a pair of connected chapter describe the relations between the spectra of Gi, ..., Gn and G. In particular, hypergraphs H, IÏ which are duals of each other (so that G is the incidence graph of H as well in some important cases, the spectrum of G is determined by the spectra of Gi,"" Gn- as of IÏ): Thus the rp-spectrum of a connected hypergraph H may be defined as the rp-spectrum of L(H) - a definition which has the advantage of being invariant under dualization. At the end of this chapter, in Section 2.6, we shall use the theory we have developed For some more results on various spectra connected with hypergraphs and graphs derived to derive the spectra and/or characteristic polynomials of several special classes of from them see (Rung). graphs. 14. A balanced incomplete block design (BIBD)t B can be considered as a special hypergraph H with the varieties and blocks of B being the vertices and hyperedges of H, respectively. So the complexity t of B may be defined as the number of spanning trees of the incidence graph 2.1. The polynomial of a graph corresponding to H. It turns out that t = t(B) is completely determined by the parameters v, b, '/, k, il of B: Let G1 = (ge, ~il and G2 = (ge, ~2) be graphst with thB (same) set of vertices t(B) = lcb-v+1ilv-1vv-2. ge = (Xi' ..., xn), where ~i and ~2 are the sets of edges of these graphs. The union (F. RUNGE (Rung); see also (RuSa)) Gi u G2 of the graphs Gi and G2 is the graph G = (ge, ~), where ~ = ~i U ~2' It is Hi. Show that the relation between the characteristic polynomial P G(il) of a graph G and the understood that every edge from ~i is different from any edge from ~2' even when characteristic polynomial SG(il) of the Seidel adjacency matrix S of G can be written in the the considered edges join the same pair of vertices. Tf Ai, A2, and A are the adjacency form matrices of graphs Gi, G2, and Gi u G2, respectively, then A = Ai + A2. p,I'i ~ I ;:io s,i~" ~-t~ )' However, Gi u G2 depends not only on Gi and G2 but also on the numeration of 1 + - HG - the vertices of these graphs. Therefore, the spectrum of the graph Gi u G2 is, in 2il il general, not determined by the spectra of Gi and G2. Some information about the where HG(t) is the generating function for the numbers of walks in G. spectrum of the union of graphs is provided by the following theorem from general (D. M. CVETKovic (CvelS)) matrix theory. t For the definition of a BIBD see Section 6.2, pp. 165/166. Theorem 2.1 (the Courant-Weyl inequalities; see, for example,' (Hofl1)): Let À1(X), ..., Àn(X) (;'I(X) ~ À2(X) ~ ... ~ Àn(X)) be the eigenvalues of a real syrnrnetric i ¡

t The "graphs" considered in this section are, in fact, multi-(di- )graphs (loops being allowed); see the general remark in the Introduction (p. 11). .

4* 3.1. Digraphs 81

3. Relations Between Spectral and Structural Properties of Graphs pletely determined by P GP'): If every vertex of G has exactly h directed loops, a¡ then h = -" - and PH(À) = PG(J, + h). n If G is a digraph without loops, then no pair of vertices of G is joined by two edges of opposite orientation if and only if a2 = O. This fact can be easily realized by considering all principal minors of the second order of the adjacency matrix. From Theorem 1.2 we deduce immediately: A digraph G contains no cycle if and only if all the coeffcients ai (i = 1, ..., n) are equal to zero, i.e., if and only if the spectrum of G contains no eigenvalue different from zero (J. SEDLÁCEK (Sed 1)). According to Theorem 1.9, the number of closed walks of given length k contained in a digraph G can be determined by means of the spectrum of G; this number is n equal to tr Ak = L Àf. In this chapter we shall describe only a part of the known relations between the i=l spectra and the structure of (multi-)(di-)graphs. These relations represent, in fact, Using the Cayley-Hamilton Theorem, we deduce from the characteristic poly- the main topic of this book, and they can be encountered in all other chapters. nomial (3.1) the following relations: As is well, known, there are some structural properties that are not uniquely determined by the spectrum, but even in these cases we can, on the basis of the An+k + a¡An+k-1 + ... + a.nAk = 0 (k = 0, 1, ...). (3.2) spectrum, frequently specify a range of variation of these properties. Therefore, By means of Theorem 1.9, we can obtain from (3.2) some information concerning many inequalities for various numerical characteristics (chromatic number, dia- the digraph structure. meter, etc.) appear in this chapter. Now we shall establish some theorems concerning the cycle structure of a digraph In all theorems of this chapter we assume that either the spectrum or the eigen- G without multiple edges. Some statements given in the foregoing- are special cases vectors of the adjacency matrix of a graph, or both, are given and that a certain class of these theorems. to which the graph belongs is specified. If the spectrum of the graph is given, we The length g(G) of a shortest cycle in a digraph G (if such a cycle exists) is called assume that its characteristic polynomial is also known, and conversely. The al- the girth of G. If G has no cycles, then g(G) = +00. Obviously, each linear directed gebraic and numerical problems which appear here are assumed to be solved. Note subgraph of G with less than 2g vertices, where g = g(G), is necessarily a cycle. that in some cases the class of graphs to which the graph with the given spectrum From Theorem 1.2 we deduce belongs can be determined by means of the spectrum. Let, as usual, A denote the adjacency matrix, let ai = LEg,L (-1 )C(Ll ë,cG = - L 1 (i .c 2g). PG(À) = IU - AI = Àn + (t¡Àn-1 +... + an (3.1) Thus -a¡ is the number of cycles of length i contained in G. be the characteristic polynomial, and p,¡, ..., Ànl the spectrum of the graph G. Theorem 3.1 (H. SACHS (Sac 3)): Let G be a digraph with the characteristic poly- nomial (3.1) and let g(G) = g. Let further i ;? min (2g - 1, n). Then the number of cycles of length i contained in G is equal to -ai' The girth g of Gis equal to the smallest 3.1. Digraphs index i for which ai =l O. This result can be extended so that the number of cycles of length i for some First we shall assume that multiple oriented edges and loops are allowed in the i ? 2g - 1 can also be ,determined. We shall introduce a new notion: the d-girth digraphs to be considered. Before formulating some theorems we shall note a few of a digraph. For an arbitrary integer d? 1, the d-girth gd(G) of a digraph G is simple facts. defined as the length of a shortest cycle among those cycles the lengths of which are The number of vertices of G is equal to the degree n of its characteristic poly- not divisible by d. If there are no such cycles, then gd(G) = +00. nomial, i.e. to the number of eigenvalues of G. The number of directed loops is equal to the trace of the adjacency matrix, i.e. Theorem 3.2 (H. SACHS (Sac 3)) : Let G be a digraph with the characteristic poly- to the sum ;.¡ + ... + Àn, i.e. to the quantity -a¡. nomial (3.1) and let g(G) = g and gd(G) = gd' Let /urtheri ;? min (g + gd - 1, n), . If every vertex of G has the same number of loops, then the characteristic poly- i =1 0 (mod d). Then the number of cycles of length i contained in G is equal to -ni' nomial PH(À) of the digraph H obtained from G by deleting all of its loops is com- The d-girth gd of G is equal to the smallest index not divisible by d for which ai =l O.

6 Cvetkovic/Doob/Sachs 82 3. Relations between spectral and structural properties of graphs 3.1. Digraphs 83

Remark. If g is not divisible by d, then, trivially, ,gd = g and Theorem 3.2 states Theorem 3.5: The characteristic polynomial of a cyclically k-partite digraph G has less than Theorem 3.1. But in the opposite case, when d is a factor of g, we certainly the form have gd ? g. If, further, gd ? g + 1, Theorem 3.2 yields new information that is not obtainable from Theorem 3.1. PG(Â) = ilP. Q(ilk), (3.3) Example. Let g = 9, gg = 15, ga = 20, Theorem 3.1 yields the numbers of cycles I where Q is a monic, Q(O) =! 0, and p is a non-negative integer. of length c for c ~ 17. In addition, with d = 9 Theorem 3.2 provides these numbers If G is a strongly connected digraph and if its characteristic polynomial is of the for c = 19, 20, 21, 22, 23, and with d = 3 also for c = 25, 26, 28. form (3.3), then G is cyclically k-partite. With d = 2 we have the following corollary. The following theorem is taken directly from the theory of matrices (see, for example, (Gant), voL. II, p. 63). Corollary: The length g2 of the shortest odd cycle in G is equal to the index of the first non-v(inishing coefficient among ai, aa, a5, ...; the number of shortest odd cycles is Theorem 3.6 : Let d-i, ..., d;; and dt, . . ., d-; be the indegrees and outdegrees, respectively, equal to -ago of the vertices of a digraph G. Then, for the index r of G, the following inequalities hold: Proof of Theorem 3.2. Let i ~ min (g + gd - 1, n), i =$ 0 (mod d). Then each linear min d¡ ~ r ~ max d¡ , (3.4) directed subgraph in G with i vertices is necessarily a cycle. As in the above argument, i i ai = ~ (_l)C(L)= - ~ 1 min dt ~ r ~ max dt . (3.5) LE!I, ë,c G i i which completes the proof. If G is strongly connected, then equality on the left-hand side or on the right-hand side of (3.4) (or of (3.5)) holds if and only if all the quantities d¡, ..., d;; (or dt, ..., d-;) are From the corollary. of Theoremí' 3.2 we can easily deduce the following theorem. equal. Theorem 3.3 (H. SACHS (Sac 3)) : A digraph G has no odd cycles ifand only if its Theorem 3.7 (A. J. HOFFMAN, M. H. McANDREW (HoMe)): For a digraph G with characteristic polynomial has the following form: the adjacency matrix A: PG().) = iln + a2iln-2 + a4),n-4 +... = ilP. Q(il2), 1° There exists a polynomial P(x) such that where Q is a polynomial and p = 0 for n even, and p = 1 otherwise. J = P(A), (3.6)

The following theorem can also easily be proved. if and only if G is strongly connected and regular. Theorem 3.4: A strongly connected digraph G with greatest eigenvalue r has no odd 2° The unique polynomial P(x) of least degree such that (3.6) is satisfied is nS(x)jS(d) cycles if and only if -r is also an eigenvalue 0t G. where (x - d) S(x) is the minimal polynomial of A (ind d is the degree of G. Proof. If G has no odd cycles then, by Theorem 3.3, -r is also an eigenvalue of G. 3° If P(x) is the polynomial of least degree such that (3.6) is satisfied, then the degree of G is the greatest real root of P(x) = n. Conversely, if -r belongs to the spectrum of G then the adjacency matrix of G is imprimitive. According to Theorem 0.3 (Section 0.3), the index of imprimitivity h Proof. Assume that (3.6) holds. Let i, j be distinct vertices of G. By (3.6), there is can in that case be only an even number. By the same theorem, there exists a per- some integer k such that Ak has a positive entry in position (i, j), i.e., there is some mutation matrix P such that PAP-I has the form (0.1). Since h is even, G obviously walk of length k from i to j. So G is strongly connected. Further, from (3.6) follows contains no odd cycles. that J commutes with A. Let ei, dj be the outdegree and the indegree of vertex i and This completes the proof. vertex j, respectively. Now the (i, j) entry of AJ is e;, and the (i, j) entry of JA is di. Thus ei = dj for all i and j, so G is regular, i.e., all row and column sums of A A digraph G is said to be cyclically k-partite if its vertex set fl can be partitioned are equal (A being not necessarily symmetric). into non-empty mutually disjoint sets fli,""fl/c so that, if (x,y) (XEfli,yEflj) To prove the converse assume G to be strongly connected and regular. Due to is an arc of G, then j - i _ 1 (mod k). Note that a cyclically k-partite digraph is the regularity, u = (1, 1, ..., l)T is an eigenvector of ' both A and AT, corresponding also cyclically l-partite if k is divisible by l. The adjacency matrix of a cyclically to the eigenvalue d. Hence, if d has multiplicity greater than 1, it must have at h-partite digraph has the form (0.1). According to (DuMe) we can formulate the least one more eigenvector associated with it. But because of the strong connected- following theorem. ness, u is the only eigenvector corresponding to d. It follows that, if R(x) is the

6* 84 3. Relations between spectral and structural properties of graphs 3.2. Graphs 85 minimal polynomial of A, and if Sex) = R(x)/(x - d), then Sed) =1 O. We then have Proof. As is well known, since the adjacency matrix A = (aij)~i of G is Hermitian, the problem of finding the maximal value of Rayleigh's quotient o = R(A) = (A -dI) S(A). (3.7) II n Let 0 be the zero-vector. Since R(A) t' = 0 for all vectors v, it follows from (3.7) that ¿ ¿ aijXiXj I R i=1 j~1 (A - dI) SeA) v = 0, n (3.10) 'V 2 l. Xi so SeA) v = IxU for some ix. i=1 Let (U, v) be the scalar product of vectors u and v. If we take (V, u) = 0, then (the Xi being arbitrary real numbers not all equal to zero) has the solution R = 1'. (AkV, u) = (v, (AT)k u) = dk(v, u) = 0 for every k and so (S(A) v, u) = O. Therefore, o = (S(A) v, u) = (IXU, u) = nix, i.e., ix = o. The maximum is attained if and only if the Xi (i = 1, ..., n) are the components of Thus SeA) v = 0 for all v such that (v, u) = 0; further, SeA) u = Sed) u. Hence an eigenvector of ~4 belonging to 1'. nS(A)/S(d) = J, i.e., (3.6) is satisfied with If we put Xi = 1 (i = 1, ..., n) in (3.10), we have n _ 1 n P(x) = - S(x). (3.8) R = d = - L di, Sed) n i=1 n This completes the proof of 1°; part 2° follows since the polynomial (3.8) has where di = L aij is the valency of vertex i. So, ìl is a particular value of Rayleigh's smaller degree than the minimal polynomial of A. To prove 3° we note that A is j~1 non-negative and has royv and column sums all equal to d. Thus, the eigenvalues of A are all of absolute value not greater than d. The roots of P(x) are eigenvalues quotient establishing (3.9). of A and hence, for real x? d, IP(x)1 is an increasing function in x. From (3.8), For regular graphs equality holds in (3.9), since in that case" the greatest eigen- P(d) = n and so, since P(x) is a real polynomiaL, P(x) ? n for x ? d. value of G is equal to the degree of G. Let, conversely, equality hold in (3.9). Theil This completes the proof of Theorem 3.7. We call (3.8) the polynomial belonging the values Xi = 1 (i = 1, ..., n) constitute an eigenvector for A belonging to r, and to G and also say that G belongs to the polynomial. n n j~1L aijxj = 1'Xi (i = 1, j~..., n) implies di1 = L aij = r (i = 1, ..., n). Thus, G is regular. (Bri1).Note that some non-regular graphs can have a polynomial with similar propertiesI This completes the proof of the theorem. Applying Theorem 3.6 to graphs and using Theorem 3.8, we get

3.2. Graphs dmin ~ d ~ l' ~ dmax'

If a multi-digraph H has a symmetric adjacency matrix A with even entries on the where dmin and dmax are the minimal and maximal values, respectively, of the valen- diagonal, then the matrix A can be understood as the adjacency matrix of an (un- cies in G. directed) (multi-)graph G. In such a way we can apply the result from Section 3.1 We continue with some more propositions relating the coefficients ai of PG(À) to to graphs. But now, due to the symmetry of the adjacency matrix, we have some some structural properties of G. further results. Due to the absence of loops, we always have a1 = O. The eigenvalues of a graph are real numbers, and we can order them so that the The number of closed walks of length 2 is obviously equal to twice the number 11 sequence Ji1, ..., Àn is non-increasing. This convention wil always be adopted. 1 ii 1 n of edges, therefore m = - L Jil. In a similar way the formula t = - L À¥ for the In the sequel we shall consider only undirected graphs without multiple edges 2 i~1 6 i=1 or loops. number t of triangles can be obtained. Now, Theorem 1.3 gives m = -ci2 and The following theorem can be proved using arguments directly from matrix theory. t = - l- 2a3. According to the' same theorem, the coefficient a4 is equal to the number Theorem 3.8 (L. COLLATZ, U. SINOGOWITZ (CoSi1)): Let ìl be the mean value of the of pairs of non-adjacent edges minus twice the number of circuits 04 of length 4 valencies and r the greatest eigenvalue of a graph G. Then contained in G. ìl ~ r, In a similar way the coefficient as is equal to twice the number of figures consisting (3.9) of a triangle and an edge (triangle and edge being disjoint) minus twice the number where equality holds if and only if G is regular. of circuits 05 of length 5. These facts were noted in (CoSi 1). 86 3. Relations between spectral and structural properties of graphs 3.2. Graphs 87

An interesting conclusion can be drawn from formula (1.36) for the coefficients of exactly two shortest odd cycles (with opposite orientations) of H and therefore the the characteristic polynomial (Sac 3). For i = n the 1-factors of G represent ony type number of shortest odd circuits in G is half the number .of the shortest odd cycles of basic figures. The contribution of a 1-factor to an is either 1 or -1, while other in H. Thus, we have the following theorem., basic- figures contribute an even number to an' Therefore, the number of 1-factors of G is congruent to an modulo 2. If an is odd, then there exists at least one 1-factor. I Theorem 3.10 (H. SACHS (Sac 3)) : Let G be a grapht with the characteristic polynomial If G is a forest then, obviously, the number of 1-factors is equal to janl with (3.1). Then the length f of a shortest odd circuit in G ,is equal to the index of the first n non-vanishing coefficient among ai, a3, a5, ... The number of shortest odd cÙ'cuits an = (-1)2 if there is a 1-factor, and an = Ootherwise.t 1 is equal to - - ai' For the proof of the following theorem we need a simple lemma which we state 2 without proof. Both, Lemma 3.1 and Theorem 3.9, wil be used in Section 7.7. An immediate consequence of this theorem is the following: Lemma 3.1: Let lXI' ..., IXk be real numbers and let 1', s (1' even, l' ~ s) be non-negative integers. Then for a :: 0 the following implication holds: Theorem 3.11: A gmph t containing at least one edge is bipartite if and only if its spectrum, considered as a set of points on the real axis, is symmetric with respect to the IX~ + ... + ix!; ~ aT =? IlXf + ... + ixti ~ CtS. zero point. Equality on the right-hand side of the implication holds if and only if the absolute value Theorem 3.11 is one of the best-known theorems making evident a close connection of exactly one of the qiintities lXI' ..., IXk is equal to a, the other quantities being all between the structure and spectra of graphs. It seems that the necessity part of eqiil to zero. Strict inequality on the left-hand side implies strict inequality on the right- this theorem was first recorded in chemical literature by C. A. COULSON, G. S. RUSH- hand side of the implication. BROOKE (CoRu) (chemists usually call it the "pairing theorem"). Theorem 3.9 (E. NOSAL (Nos 1)): Let (J,i, ..., ,1n) be the spectrim of a graph G. Then The entire theorem was proved by H. SACHS (Sac 7) in the form of Theorem 3.3. , he inequality It is of interest that this theorem has been rediscovered several times. Various versions of the theorem can be found in (CoSi1), (Hof3), (Cve1), (CoLo), (Rou1), (Mari), (Sac3). ;,i :: ,1~ + ,1~ + .., + ;,~ (3.11) The characterization of connected bipartite graphs by Theorem 3.4 is also possible. implies that G contains at least one triangle. We shall now consider the problem of determining the girth of a graph. As in di- graphs, the girth of a graph G is the length of the shortest circuits of G. Proof. According to Lemma 3.1, (3.11) implies If we try to formulate a theorem similar to Theorem 3.1 for graphs, we encounter the following difficulties: Together with the graph G, consider the digraph H which has the same adjacency matrix as G. If G contains at least one edge, then g(H) = 2, I it2,1T I ~ ,1i while g(G) can at the same time be arbitrarily large. Thus the girths of G and Hare and we obtain for the number t of triangles not related. 1 3 1 ~ 3 1 3 1 I.~ 31 0 But it is easy to see the following. For i ~ g( G) there exist basic figures only for t = - ,11 + - ~ ;,¡ ~ - ;'1 - - kJ ,1i :: . i = 2q even, and each basic figure U2q consists of q non-adjacent edges, so that 6 6 i~2 6 6 i=2 p(U2q) = q and c(U2q) = O. Therefore, This completes the proof.

11 ai = (i ~ g(G)), Since ¿ ,1f = 2m, where m is the number of edges of G, we get the following r. (-0 l)q for bq forodd i = 2qi ¡~l corollary. where bq is the number of basic figures consisting of exactly q non-adjacent edges. For i = g(G) basic figures can be of the described type (consisting of non-adjacent Corollary: If ;'1 :: Vm, then G contains at least one triangle. edges; only for even i), or they can be circuits of length g(G). In the second case the The corollary of Theorem 3.2 can be reformulated for (undirected) graphs in the contribution of each such basic figure to ai is -2. If following way. Let us consider, together with a graph G, the digraph H which has the same adjacency matrix as G. To each shortest odd circuit of G there correspond I ai = 1J a¡ a¡ - (- for l)q bqodd for i =i, 2q, i J t From (1.35) foIlows: The number of directed 1-factors (= linear directed subgraphs with n vertices) of any digraph G is not smaller than Ian!' I t Theorems 3.10, 3.11 hold for multigraphs, too

I 88 3. Relations between spectral and structural properties of graphs 3.2. Gra,phs 89

then ai = 0 for i ~ g(G) and -ag(G) is equal to twice the number of circuits of where p_, Po, p+ denote the number of eigenvalues of G smaller than, equal to, 01' greater length g(G). than zero, respectively. So we have the following There are gmphs for which equality holds in (3.13). Theorem 3.12 (H. SACHS (Sac3)): Let G be a (multi-)graph with the characteristic Proof. Let s = Po + min (p_, p+). Suppose that there is a graph G for which

polynomial (3.1) and let bq be the number of basic figures consisting of exactly q non- Ix(G) / s holds. Thcn there is an induced sub graph of G with Ix(G) vertices containing adjacent edges. Let further no edges. Thus, a principal submatrix, of the order Ix(G), of the adjacency matrix of G is equal to the zero-matrix. Since all eigenvalues of a zero-matrix are equal to _ i for odd i, zero, Theorem 0,10 gives for the eigenvalues )'1' ..., ),,, of G the inequalities ai = ai _ (-l)q bq

fa' for i = 2q. ).i ~ 0, À,,-o(G)+i ~ 0 (i = 1, ..., Ix(G)). Then g(G) is equal to the index of the first non-vanishing numbe1' among lI, a2, ..., However, this contradicts the assumption Ix(G) / s. Thus, (3.13) holds. and the number of circuits of length g(G) is equal to - l- ag(G)' Equaliy holds in (3.13), for example, for complete graphs. (For regular graphs see also Theorems 3.26,3.27.) 2 This completes the proof of the Theorem 3.14. A special case of this result is noted in (Bax 1) This paper deals with an adjacency Since the adjacency matrix A of a graph is symmetric, we can determine its mini- matrix of a somewhat different structure. mal polynomial on the basis of the spectrum. As is well known, if rÀ(l), ..., ),(m)) is the set of all distinct eigenvalues of A, the corresponding minimal polynomial qi(íl) is Theorem 3.15 (See (Cve9), (Cve12), (AmRa), (Rof16)): Let P=-i' P_i, andp~i denote given by the number of eigenvalues of the graph G which are smaller than, equ(il to, or greater

than -1. Let ),* represent the smallest eigenvalue greater than -1. Let further p = P=-l qi(À) = (À -_ íi)) ... (À - íl(m)). + P_i + 1 and s = min (p, P~i + P_i, l' + 1), where i' is the index (= maximum eigen- Let qi().) = Àm + blÀm-i +... + bm. Then the following relations hold: value) of G, and let Am+k + bIAm+k-l + ... + bmAk = 0 (1: = 0, 1, ...). (3.12) ix=f~ if ),* ~ p - 1, Using these relations we can prove the following theorem ((Nos1), (Cve9); see also,i if íl * / P - 1. for example, (MaMi), p. 123). If K(G) denotes the 'maximum number of vertices in a complete 8ubgraph of G, then Theorem 3.13: If a, connected graph G has exactly m distinct eigenvalues, then its if s ~p, diameter D satisfies the inequality D ~ m - 1. K(G) ~ J s (3.14) ls-iX if s = p. Proof. Assume the theorem to be false. Then for some connected graph G we have There are graphs for which eq'uality holds in (3.14). D = s ~ m. By the definition of the diameter, for some i and j the elements aW) from the i-th row and from the j-th column of the matrices Ak (k = 1, 2, ...) are Proof. If G contains a complete subgraph with k vertices, then Theorem 0.10, equal to zero for k ~ s, whereas aW =F O. in a way similar to the proof of Theorem 3.14, yields the following inequalities: In (3.12), put k = s - m. Making use of the relation so obtained, from aW = 0 Àn-k+1 ~ 1: - 1 ~ )'1 = 1', (k = 1, ..., s - 1) we deduce aW = 0, which is a contradiction. This completes the proof of the theorem. ' Àii-k+i ~ -1 ~ )'i (i = 2, ..., k). The interior stabilty number (\(G) of the graph G is defined as the maximum The greatest value of k satisfying these inequalities is given by the expression on number of vertices which can be chosen in G so that no pair of them is joined by an the right-hand side of inequality (3.14). Equality holds in (3.14), for example, for com- edge of G. . plete multipartite graphs. This completes the proof of Theorem 3.15.

Theorem 3.14 (D. M. CVETKovic (Cve9), (Cve 12)): The interior stability number In the paper (JIux 1) the author deals, among other things, with the connection ix( G) of the graph G satisfies the inequality between the spectrum of a graph and the maximum number of vertices in a com- plete subgraph. The proofs of the results announced in (JIux1) have, ,as far as we Ix(G) ~ Po + min (p_, p+), (3.13) know, not yet been published. 90 3, Relations between spectral and structural properties of graphs 3.2. Graphs 91

Now we shall discuss the relations between the spectrum of a graph and its chro- It can be proved (see (LiWh)) that every graph G contains an induced subgraph H matic number. It is surprising that on the basis of the spectrum, some information with dmin(H) ~ (k + 1) (Qk(G) - 1) On the basis of this fact the following theorem about the chromatic number (a quantity which in general cannot easily be deter- (Lick) can be proved in a manner similar to the last one (for the special case k = 1, mined) can be obtained. For some special classes of graphs the chromatic number can see also (Mi tc )). even be calculated exactly from the spectrum (for example, for bichromatic l graphs: see Theorem 3.11; for regular graphs of degree n - 3, where n is the number Theorem 3.17 (D. R. LICK (Lick)): For any gralJh G with inde;r r wid (my non- of vertices: see Section 3.6). Nevertheless, in the majority of cases, we have some negative integer k, inequalities for the chromatic number. In general, these inequalities are not too sharp, but for each inequality there are graphs for which this inequality yields a Qk(G) S 1 + r~J. good (lower or upper) estimate of the chromatic number. Therefore, all known esti- - Lk + 1 mates should be applied to the given graph, and then the best one should be For the proof of the following theorem we quote a lemma without proof; both the chosen. lemma and theorem have been proved in (Hof16). In general, however, the chromatic number is not determined by the spectrum. Moreover, A. J. HOFFMAN has proved that there is, in a certain sense, an essential L.emma 3.2: Let A be a real symmetricnwtrix of order n, and let Yi u .,. u Yt irrelevance between the spectrum and the chromatic number of a graph (see Sec- (t ~ 2) be a partition of 11, ..., nl into non-empty subsets. Akk denotes the submatrix tion 6.1). of A with row and column indices from Yk' If 0 ;; ik ;; IYkl, k = 1, ..., t, then We shall now present some theorems concerning the topic under consideration. We begin with a theorem due to H. S. WILF. t-I t lii+i,+...+i+I(A) +i=1 ~ In-i+1(A) k=1 ;; ~ lik+1(Akk), (3.17) Theorem 3.16 (H. S. WILF (Wil2J): Let X(G) be the chromatic number and r ~he index (= maximum eigenvalue) of a connected graph G. Then where Ài(X), i = 1, 2, ..., are the eigenva.lues of the matrix X in decreasing order.

x(G) ;; r + 1. (3.15) Theorem 3.18 (A. J. HOFFMAN (Hof16)): If r (r =f 0) and q (ire the greatest and the smallest eigenvalues of the graph G, then its chTomatic numbe'l' x(G) satisfies the in- Equ(ility holds if and only if G is (i complete graph or a circuit of odd length. equality

Proof. Let dmin(H) and dmax(H) denote the smallest and the greatest vertex degree l' x( G) ~ - + 1. (3.18) in a graph H and let ll(H) be the index of H. Since X(G) is the chromatic number -q of G, there exists an induced subgraph H of G with dmin(H) ~ X(G) - 1. By Theo- rems 0.6 and 3.8 we obtain Proof. Let X(G) = t and let the vertices of G be labelled by 1, ..., n. Then there . existj, a partition Yi u ... u Yt such that each of the subgraphs of G induced by Yi ll(G) ~ i.I(H) ~ dmin(H) ~ X(G) - 1, (3.16) contains no edges. With ik = 0 (k = 1, ..., t), (3.17) yields for the eigenvalues and therefrom (3.15). Let equality now hold in (3.15) and, consequently, also in )'1 = 1', ,12' ..., 171 = q of G

(3.16). Then ¡'i (G) = ll(H) implies G = H, since G is connected. Further ll(G) t-I = dmin(G), which implies, according to Theorem 3.8, that G is regular. Thus i' + ~,1n-i+l;; O. (3.19) i=1 X(G) = 1 + l' = 1 + dmax(G). The well-known Brooks Theorem (see, e.g., (Sac9)) . now implies that G is a complete graph or a circuit of odd length. This completes t-I Since ~ In-i+1 ~ (t - 1) q, (3.18) follows from (3.19). This completes the proof of the proof. i~1 Before we quote a generalization of this theorem we shall give some definitions. the theorem. A graph G is k-degenerate, for some non-negative integer k, if dmin(H) ;; k for each Note that from (3.19) there can be drawn more information about the chromatic induced subgraph H of G. The point partition number ek(G) of the graph G is the number than from (3.18). Actually, (3.19) yields the following bound: smallest number of sets into which the vertex set of G can be partitioned so that each set induces a k-degenerate subgraph of G. Since O-degenerate graphs are exactly X(G) ~ 1 + min Ix I ,11 + ~ ,1n-i+1 ;; ol. those which are totally disconnected, we see that eo(G) is the chromatic number l( 1 i=1 J of G. Qi(G) is called the point arboricity of G, since 1-degenerate graphs are forests. The following theorem provides another lower bound for the chromatic number. 92 3. Relations between spectral and structural properties of graphs 3.2. Graphs 93

Theorem 3.19 (D. M. CVETKOvrC (Cvell)): If G is (¿ graph with n ve1.tices, with Theorem 3.20 (A. J. HOFFMAN, L. HOWES (HoHo)): Let m(G) be the number of index r and chromatic number X(G), then the following inequality holds eigenvalues of a graph G not greater than - 1. Then there exists a function f such that i X(G) ~ f(m(G)). .

x(G)~~n - r' 1 Proof. Let e = e(G) be the largest number such that G contains a set of 2e vertices 1, ..., e, 1', ..., e' with i and i' adjacent (i = 1, ..., e), other pairs of vertices being Proof. Consider the characteristic polynomial of a k-complete. graph K"".."u" not adjacent. Let K(G) be the maximum number of vertices in a complete sub- which is given by (2.50) or. (2.51). The polynomial (2.50) has a single positive root graph of G. Using Theorem 0.10 we simply obtain K(G) ~ 1 + m(G), e(G) ~ m(G) which is simple. Indeed, as is seen in Theorem 6.7, complete multipartite graphs are (see also Theorem 3.15). Hence, we have only to prove that X(G) is bounded by some precisely those connected graphs with a single positive eigenvalue. Thus for function of K(G) and e(G). This wil be done by induction on K(G). If K(G) = 1, x:: 0, PK (x) ~ 0 if and only if x ~ 1.1' ni'....nk then X(G) = 1. For a given graph G, let G¡ (O¡,), i = 1, ..., e, be the subgraph of G Now consider the values of induced by the set of vertices adjacent to i (i/). Since K(G;), K(G;,) -e K(G) and e(G;), e(Gi,) ~ e(G), the induction hypothesis can be applied to Gi (G;,). But the set Ie Ie of vertices of G contained in no G; or Gi, induces a subgraph without edges. This L~ Ln; =n. (3.20) ;=11. + n'i ' 1.:: 0, ¡~1 fact is sufficient to prove the theorem. Assunie for the moment that the n;'s can assume positive real values. Then (3.20) It was conjectured in (Ho!16) that f(m(G)) = 1 + m(G). But as was observed in attains its maximum when all the n;'s are equal. Indeed, if n¡ =1 ni' then by letting (HoHo), this is false since C 7, the complement of a circuit of length 7, provides a ni = nj = l- (ni + nil and by leaving all other values unchanged, (3.20) is in- counter example. In (AmHa) it was mentioned that the inequality X(G) ~ P + 1, 2 k' t where p is the number of non-positive eigenvalues of G, might possibly be valid. It creased. For the particular value 1. = ~ n, (3.20) is equal to 1 when the n;'s are can be shown that at least one of the inequalities X(G) ~ rk (G), X(G) ~ rk (0) k equal. Thus when the n;'s are positive integers (2.50) is non-negative and hence (rk (G) being the rank of the adjacency matrix of G) holds (Nuf2). On the basis of this and some other facts, it can be conjectured that, except for Kn, X(G) ~ rk (G), k -1 where equality holds if and only if the non-isolated vertices of G form a complete íl~-n1 = k (3.21) multipartite graph (Nuf2). Some other bounds for the chromatic number wil be given in Sections 3.3 and 3.6. with equalìty only when the graph is regular. So we have proved Now we shall give some bounds for certain quantities connected with the par- titioning of the edges of a graph G. Lemma 3.3: The index r of K"".."u. satisfies Let b( G) be the smallest integer k such that there exists a partition

k - 1 k Æ'i u... u Clk = qt, (i, j = 1,2, ..., k; i =1 j) (3.22) r ~ - kn where ;=1 n = L ni' Æingi=0 of the set úl of edges of G and such that the subgraph G; of G, induced by g;, is a (In the Appendix the spectra of some k-complete graphs are given.) complete graph for each i = 1, ..., k. Let £(G) be the smallest integer k such that H X(G) = k, the set of vertices of G can be partitioned into k non-empty subsets so that the subgraph induced by anyone of these subsets contains no edges. If the (3.22) holds and each G; is a complete iiultipartite graph. Further, let #(G) be the smallest integer k such that (3.22) holds and each G; is a bicomplete graph. mentioned subsets contain n1,..., nk (n1 + ... + nk = n) vertices, respectivlIy, graph then by adding new edges to G we'can obtain Kn"....Uk It is known (see Theorem 0.7) Theorem 3.21 (A~ J. HOFFMAN (Hof 11)): Let 1.1, ..., J,u be the eigenvalues of a G. tJiat the index of a graph does not decrease when new edges are added to the graph. Let p+, p_, P be the number of eigenvalues which are positive, negative, different from Therefore, the index of G is not greater than the index of KU",..,uk According to both -1 and 0, respectively. Then this and the foregoing. r ~ k - 1 n, which implies k ¿ ~ . , - k -n-r £(G) ~ p+, #(G) ~ p_, b(G) ~ -Jon, r + 2å(G)) ~ p. This completes the proof of the theorem. The following theorem of HOFFMAN and HOWES esta,blishes the existence of an This theorem was proved by means of the Courant-Weyl inequalities (see Theo- upper bound of another type for the chromatic number. rem 2.1). In the proof several lemmata appear. They are included in Section 3.6. 94 3. Relations between spectral and structural properties of graphs 3.3 Regular Graphs 95

The essential irrelevance (in a sense) of the graph spectrum with respect to å(G) Proof. Let aij be the elements of the adjacency matrix. Then a2 is given by and 8(G) has also been shown in (Hofll); fJ(G), however, is closely related to the spectrum of G. a2 '\"- i 0-" aij .. 1- g-- l- (¿,¡~ ~ ?, . kj aji 0 2 i=1 j=1

l If Zij is the number of circuits O2 containing the vertices i and j, then Zij = (a~j) and 3.3. Reguar graphs we get 1 n n n n In the theory of regular graphs, numerous new theorems are valid that do not hold 4z = 4. - )'"- "z,. g- = " '1~ (a?, g-.. - a',) ,¡= -2a2 il - nr. for non-regular graphs. Naturally, all theorems of section 3.2 hold also for regular 2 i=1 j=1 i=1 j=1 . graphs. Corollary: G has no multiple edges if and only 1:j 2a2 = -nr.

We shall start with the question: How can it be decided by means of its spectrum Since for graphs the minimal polynomial is obtainable from the spectrum, Theo- whether or not a given graph is regular? rem 3.7 takes now the following form. .

Theorem 3.22: Let )'1 = r, )'2' ..., )'n be the spectrum of a graph G, T being the index Theorem 3.25 (A. J. HOFFMAN (Hof3)): For (i graph G with adjacency matrix A of G. G is Tegular if and only if there exists a polynomial P(x), such that P(A) = J, if and only if G is regular and con- nected. In this case we have -1 ~/''.l ~ ,2 = T. n ;=1 (3.23) P(x)n(x - ),(2)) ...- (x - Â(m)), (1' - Â(2)) ... (r - Â(m)) If (3.23) holds, then G is regular of degTee r. where n is the number of vertices, l' is the index, and Â(I) = r, ),(2), ..., ),(m) are all distinct _PToof. 2m Sincein the mean value d of the vertex degrees in G is given by eigenvalues of G. d = - = - ~ Â~ (m is the number of edges), Theorem 3.22 is a corollary of n n i=1 This important theorem provides great possibilities for the investigation of the Theorem 3.8. structure of graphs by means of spectra. It wil be used many times in the sequel. This theorem is implicitely contained in (CoSil). See also (Cve7). It can be easily We proceed now to the investigation of the circuit structure of regular graphs. vVe modified for the case when the existence of loops in some vertices of G is allowed. shall apply Theorem 3.12 and a result from (Sac4). Consider a regular graph G. If G contains multiple edges or multiple loops, Theorem 3.33 can be applied for the According to (Sac4), in regular graphs the number bq occurring in Theorem 3.12 can establishment of regularity. for q -: g(G) be expressed in terms of q, the number of vertices n, and the degree 1" The following theorem is obvious. of G. Since nand r are obtainable from the characteristic polynomial 'of G, the fol- iowing result is immediately obtained. Theorem 3.23: The numbeT of components of a Tegular graph G is equal to the multi- Theorem 3.26 (H. SACHS (Sac 3)) : The girth g and the number of circuits of length g plicity of its index. of a regular graph G are determined by the corresponding characteristic polynomial Theorems 3.22 and 3.23 wil be used several times in this book. In many theorems PG(Â). a graph G is required to be 1° regular, or 2° regular and connected. These conditions N ow.we can go further and extend the whole of Theorem 3.12 to the case of regular can be replaced by the following ones: 1° The spectrum of G satisfies (3.23), 2° The graphs. We shall again use a result from (Sac4). spectrum of G satisfies (3.23), and r is a simple eigenvalue. Thus, in such theorems Consider the basic figures Ui with i (g ~ i -: 2g) vertices contained in G. Let U~ the assumptions concerning the general graph structure are only seemingly of a non- be those basic figures which contain no circuits (i.e. which contain only graphs K2 spectral nature. as components) and let Ai be their number. For odd i there are obviously no basic The following theorem is taken from (Fine). figures U~. For i = 2q we have Ai = bq (numbers bq b~ing defined in Theorem 3.12), Theorem 3.24 (H.-J. FINCK (Fine)): Let z be il¡e number of ciTcuits O2 of length 2 in and the contribution of U~ to the corresponding coefficient (- l)i ai of the charac- a Tegular multi-graph G of degTee T with n veTtices and without loops. Then 4z = -2a2 teristic polynomial of G is (- 1 )q\= (- 1) +. Let us further consider those basic figures - nT, wheTe a2 is the coefficient of Ân-2 in the characteristic polynomial of G. U~ which contain a circuit of length c. Clearly, g ~ c ~ i; U¡ contains exactly one 96 3. Relations between spectral and structural properties of graphs 3.3. Regular graphs 97 circuit, since, by hypothesis, the number i of vertices of U¡ is smaller than 2g. Now, respect to the desired numbers D¡. If, for example, g is even, using Theorem 3.12 we i - c vertices, not belonging to that circuit, are vertices belonging to i - c graphs obtain in order from (3.25): 2 1 _ K2; thus, c = i (mod 2) must be valid. The contribution of a basic figure' U~ to D g+1- -- 2ag+1' ( -l)i a¡ is, according to Theorem 1.3, equal to i-c i+c from (3.24): - 1+- 1 (-1) 2 .(-1)C+1.2=2.(-1) 2 Dg+2 - - 2 (ãg+2 - 2Eg+2.gDg), The last formula holds also in the case c = i, when Uj reduces to a circuit of length i. Now, for each c with g ;; c ;; .i, i _ c (mod 2) the number B¡ of different basic where, according to Theorem 3.26, Dg is the known number of circuits of length g, figures U¡ must be determined. from (3.25): Let a have exactly Dc circuits of length c and let these circuits be denoted by 1 Dg+3 = - 2 (ãg+3 - 2Eg+3,g+1Dg+1), C1 (j = 1, ..., Dc)' If c = i, we have Bj = Di, while in the case c ~ i we have the following situation: Let a1 be the subgraph of a induced by those vertices not lying on C1. Then the etc. The numbers ãj defined in Theorem 3.12 are, as has already been stated, deter- number of basic figures U¡ which contain a fixed,circuit C~ is obviously equal to the mined by 1', n, and aj, ~.e. by Pe(J,). Thus, we have proved the following theorem. number E~.c of forests, containing exactly i ~ c graphs K2 as components, in a1. Theorem 3.27 (H. SACHS (Sac3)): Let a be a regular graph with gÙth g and with According to (Sac4) the number EL depends only on i, n, 1', and c, but neither the characteristic polynomial (3.1). Let h ;; n be a non-negative integer not greater than on j nor on the special structure of a. Therefore we can omit the upper index j and, 2g - 1. Then the number of circuits of length h, which are contained in a, is determined since the numbers nand l' are directly obtainable from Pe(J..), we can assume that by the largest 1'00t r and the first h coefficients ai, a2, ..., a~ of the characteristic poly- the numbers Ei.c are also given through Pe(J,). nomial of a. So we have for c ~ i The following theorem establishes a spectral property of self-complementary D, Bi = L E~,c = Ei,cDc' graphs. A graph a is self-complementary if it is isomorphic to its complement G. i=1 Self-complementary graphs were primarily studied by G. RINGEL (Ring) and H. SACHS (Sac1) (see also (Clap), (Rea 1)). If a is a regular self-complementary If b(Ui) is' the contribution of the basic figure Ui to the corresponding coefficient gi'aph, then a is connected and has n = 4k + 1 vertices and degree r = 2k (Ring), (-I)iai of Pe(J,), we obtain (Sac1). We shall assume n? 1, i.e. k ~ 1. According to Theorem 2.6, (-l) ai = L b(U?) + L L b(U~) + L b(Uj) Å - 2k Pe(-Å _ 1), gS;c~i Uc PG(Å) = Pe(J,) = - À + 2k + 1 u?i c ~ i (mod2) i uii and hence for even i or -i 1+-i+c Pe(J,) Pe(-À - 1) ai = (-1) 2 b ¡ + L (-1) 2 2Ei,cDc - 2Di, (3.24) Å - 2k -Å - 1 - 2k "2 !JS;c~i c~O (mod2) If Åi (i = 2, 3, ..., 4k + 1) are the eigenvalues of a different from the index )'1 (i.e., and for odd i )'i =f r = 2k), then i+c 4k+l 4k+l 41c+l ai= L (-1)22Ei.cDc-2Di. (3.25) n (Å - J'i) = n (-Å - 1 - Ài) = I1 (Å + 1 + Àj). g~c.ci i~2 i~2 j-2 c~1 (mod2) To each eigenvalue Î'i =f 2k there corresponds another eigenvalue )'j = -Ài - 1, These formulas hold if i is smaller than 2g. where Åj =f Åi, since otherwise Åi = - l- which is impossible due to the fact that Åi By a recursive procedure, equations (3.24) and (3.25) can easily be solved with 2

7 Cvetkovic/Doob/Sachs 98 3. Relations between spectral and structural properties of graphs 3.3. Regular graphs 99

Let us first consider lower bounds. Obviously, X(G¡ V G2) = X(G¡) + X(G2). is an algebraic integer. Thus, .1i+1 ? - 1- and .1j = .1n+1-i1 ~ - - fori = 1,2,..., 2k, 2 2' Therefore, and x(G) ~ k (3.27) \ (3.26) )'i+1 + .14k+2-i = -1 (i = 1,2, ..., 2k), I if G can be represented as a v-product of k v-prime factors. i giving rise to the following theorem. Let G be a connected regular graph of degree l' with n vertices, n ? l' + 1 (com- plete graphs are thereby excluded, but this limitation is not essential). The multi- Theorem 3.28 (H. SACHS (Sac1)): The characteristic polynomial of a regular self- plicity of an eigenvalue .1 of G wil be denoted by Pi.' complementary graph has the form Let Pr-n + 1 = k ? 1. According to Theorem 3.29, G can be represented as a Ü+l Ü v-product of k v-prime graphs, say, G. (v = 1,2, ..., k). Each of the G. is regular PG(.1) = (), - 2k) IT (.1 - )'i) (.1 + .1i + 1) = (.1 - 2k) IT (.12 + .1 - IXi), and its degree r. and number n. of vertices satisfy the equation n. - r. = n - l' (see i~2 i~l Section 2.2, p. 57). where IXi = .1T + ¡ + )'i+1' Suppose that among the k v-prime factors of G there àre exactly m¡ monochro- ~ote that this theorem also follows from Theorem 2.10 (p. 59); see also the foot- matic, i.e. totally disconnected graphs. For such factors G., X(G.) = 1, and for the note to p. 57. other k - m¡ factors G., x(G.) ~ 2. So, Formula (3.26) implies )'2 ~ 2k - 1, since in the opposite case we should obtain x(G) ~ m¡ + 2(k - m¡) = 2k - mi' (3.28) the impossible relation )'4k+1 ~ -(2k - 1) - 1 = -1' (note that a self-complementary graph with n ? 4 cannot be bipartite, thus .14k+1 ? -1'; see Theorem 3.11). Assume further that exactly m2 of the factors G. are bichromatic, i.e. bipartite of The converse of Theorem 3.28 does not hold. Namely, there are connected regular positive degrees 1'., and let m = 2m¡ + m2' Then . graphs with 4k + 1 vertices which have the characteristic polynomial of Theorem 3.28 and which are not self-complementary. Such graphs wil be mentioned in Chapter 6 x(G) ~ m¡ + 2m2 + 3(k - m¡ - m2) = 3k - m. (3.29) (see examples of cospectral pairs of graphs consisting of a graph G and of its com- So, any upper bound for m¡ or m wil, by virtue of (3.28) or (3.29), automatically plement a). yield a (possibly trivial) lower bound for X(G). i A statement ,similar to Theorem 3.28 can be made for non-regular self-comple- Before we can outline a method of finding upper bounds for 1n¡ or m, we need mentary graphs. As a simple consequence of Theorem 2.5, we obtain the following two more definitions. statement (Cve8), (Cve9): ').0 Let (.1~, .1~, ..., ),;,,) be the spectrum of a graph G'. Then the family (.1~,.1;, ..., .1~,J Let G be a self-complementa1'Y graph. Then to each eigenvcilue )'i of G of multiplicity is called the reduced spectmm of GJ. p ? 1 (if the1'e'is such an eigenvalue) there corresponds another eigenvalue .1¡ whose 20 Let ff¡, ff2, ..., ff. be subfamilies of a finite family ffo and let pu(e) be the multi- multiplicity q satisfies the inequality p - 1 ~ q ~ p + 1, where .1i + .1j = - 1. plicity (possibly zero) with which element e is contained inff u (0- = 0, 1, ..., s). We shall now discuss some theorems which are closely connected with the concept , . ff1, ff2, ..., ff. are called inclependent in ffo if ~ pu(e) ~ po(e) for each element e of the V -product of graphs introduced in Section 2.2. A graph is called V -prime if it cannot be represented as a v-product of two graphs. of ffo' . u~¡ Now, according to Theorem 2.9, the reduced spectra of the v-prime factors G. of Theorem 3.29 (H.-J. FINCK, G. GROHMANN (FiGr)): Let G be a regula1' connected G constitute a family of independent subfamilies of the spectrum of G. Evaluating graph of degree l' with 1' vertices. G can be 1'ep1'esented as a v-product of p + 1 (p ~ 0) the conditions which the spectra of totally disconnected and regiilar bipartite graphs V -prime graphs if and only if l' - n is a p-fold eigenvalue of G. must satisfy, we can in principle easily obtain upper bounds for m¡ and m. Suppose G* to be a totally disconnected V -prime factor of G. Then 1'* = 0 and Proof. G can be represented as a v-product of p + 1 V-prime factors if and only n* = n - r. The reduced spectrum of G consists of n - l' - 1 ? 0 numbers all if a has p + 1 components. According to Theorem 3.23, this situation arises if and equal to zero: so, zero is contained in the spectrum of G with a multiplicity not only if the graph a, whose index is ;¡ = n - l' - 1, has the number n - l' - 1 as a smaller than m¡(n - l' - 1). Thus Po ~ m¡(n - l' - 1) and, consequently, (p + I)-fold eigenvalue. By virtue of Theorem 2.6, the last statement is equivalent to the statement that l' - n is a p-fold eigenvalue of G. This completes the proof. m¡ ~ ( Po J . (3.30) This theorem enables us to calculate several lower and upper bounds for the n-r-1 chromatic number X(G) of regular graphs G which are not V-prime. (3.30) is a fortiori true if m¡ = O.

7* 100 3. Relations between spectral and structural properties of graphs 3.3. Regular graphs 101

Recall that k = P.r-n + 1. From (3.28) we deduce g") Y u i2J is identical with the family of roots of the equation 1 f!(À) = 0 (see (3.32)) for some partition fl of n - l' + 2 into even numbers ~ 4. x( G) ~ 2p.r-n + 2 - ( Po J . (3.31) n-1' - 1 Note that conditions c)"-e) are consequences of conditions I'), g') or f"), g"), respectively. I A better estimate may be obtained by taking possible bichromatic v-prime factors Now let S be a family of Ui + U2 ~ k independent subfamilies Y of the spectrum into consideration. Denote by Pl the set of regular bipartite V-prime factors of G of G, exactly Ui of them having zero as their only element with multiplicity n - l' - 1, having positive degrees: then IPlI = m;. For every G, E Pl, 1', ~ ~ n, (note that a and each of the remaining U2 families satisfying the conditions given above for some i; such a family S wil be called feasible. There is, in particular, a feasible S = S* regular bipartite graph of positive degree has an even number of vertices); moreover, (with Ui = u~, U2 = u~) 'which is identical with the family of the reduced spectra of 1', ~ ~ n, - 1, because a regular bipartite graph G' with 1" = ~ n' is bicomplete all monochromatic andbichromatic v-prime factors of G, thus u~ = mi, u~ = m2, 2u~ + u~ = m. Consequently, the maximum value M of 2ui + U2, taken over all and therefore not v-prime. The above inequality, together with the relation n, - 1', feasible S, is an upper bound for m, and so = n - 1', implies 1', ~ n - l' - 2. An arbitrary subfamily Y = (,ui, ,u2, ...,,un -iJ of the spectrum of G can be the x(G) ~ 3k - M. g' reduced spectrum of a regular bipartite V -prime graph G, E Pl of some degree L. L. KRAUS and D. M. CVETKOVIÓ (KrCl) noted that all constraints to be satis- 1', = i ? 0 only if the following conditions be satisfied: fied (in particular the condition of independence) can be given the form of linear inequalities so that M may be obtained as the solution of an integer linear pro- a) n - l' + 1 ~ ng' = n - l' + i ~ 2(n - l' - 1); gramming problem which we shall now formulate. b) ng' is even; Let Yi be the family having zero as its only element, with multiplicity n - r - i. c) -i ~,ui ~ i (l = 1,2, ..., ng' - 1); Determine the set iY2, Y3, ..., Y/J of all distinct (not necessarily independent) d) the fàmily Y u iiJ = (i, ,ui, ,u2, ..., ,un g'-i) is symmetric with respect to the zero subfamilies Y of the spectrum of G satisfying, for some i, the conditions given above. Let the spectrum of G contain the distinct eigenvalues ),(i) (i = 1,2, ..., d) with point of the real axis (,u and -,u in Y u i iJ having the same multiplicity); multiplicities Pi (Pi + P2 + ... + Pd = n). Let Pij be the multiplicity in Yj of the n g'-i eigenvalue ),(i). If Yj appears e:xactly Xj times as an element of the feasible family e) ~,uf = i(n - 1'). J i=i S then, with the notation used above: Xi = Ui, ~ :"Cj = U2, and so the following These conditions are either direct consequences of Theorems 3.11,2.9,3.22 or obvious. inequalities hold: j~2 In the cases i = 1 and i = 2 stronger conditions that Y must satisfy can be formulated. Xj ~ 0 (j = 1, 2, ..., f) , (3.33)

Case i = 1: In this case, G, has only complete. graphs K2 as components. I ~ Xj ~ k, (3.34) f') ng' = n - r + 1 ~ 4 (since K2 is not V-prime); j=i .r ~ 1 n-1'-l ~PijXj ~ Pi (i = 1,2, ..., d); , (3.35) g') Y contains number 1 with multiplicity2 - n g'2 - 1 = and number -1 j=i wit. hmu 1 tip . icity1" 1- n ng' =- .r an + no 1 ot . erd num h bers. the last inequality is equivalent to the independence of the familes Y of S as 2 2 I Case i = 2: In this case, G, has only circuits of even length ~ 4 as components. subfamilies of the spectrum of G. Since 2ui + U2 = 2xi + ~ Xj' it is clear that the The characteristic polynomial is of the form I j=2 maximum value of 2xi + ~ Xj' where the Xj are integers subject to the contraints a ( 2:7l) j~2 ff!(À) = IT .I À - 2 cos -- , (3.32) (3.33)-(3.35), is equal to the maximum value M of 2ui + U2' aEf! i~i IX SO we have proved where fl may be any partition of n - l' + 2 into even numbers ~ 4, and IX runs Theorem 3.30 (L. L. KRAUS, D. M. CVETKOVIÓ (KrCl)): Under the assumptions through all elements of fl (see end of Section 2.1). I made above, let JJI be the maximum value 0/2xi + ~ Xj' where the Xj (j = 1,2, ..., I) f") ny = n - r + 2 ~ 6 (since a circuit of length 4 is not V-prime); j=2 102 3. Relations between spectral and structural properties of graphs 3.4. Some remarks on strongly regular graphs 103 are integers subject to the constraints (3.33)-(3.35). Then In summarizing we obtain x( G) ;S 3k - M, where k = Pr-n + 1. (3.36) s ;: + P2 + - , Along similar lines, some rougher (but more easily calculable) 100.;er bounds have (Pon-r-l J (p-iJ 3 been obtained by H.-J. FINCK (Finc) and again by L. L. KRAUS and D. M. CVETKOVIC I " and, consequently, (KrC1). Now we proceed to the determination of an upper bound for the chromatic number (3.39) of a regular graph which is not V -prime. X( G) ;: kr - (k - 1)n n,- + r + - P2 1 +3 - . Let, as earlier, G be a connected regular graph of degree r with n vertices and let , (po J (p-i) the eigenvalue l' - n occur in the spectrum of G with multiplicity k - 1 (;S 0). Having in view relations (3.38) and (3.39), we can formulate the following theorem. Then the v-prime factors of G are regular graphs G. of degree rv with nv vertices, where r. = l' - n + n. (v = 1, ..., k). According to the well-known theorem of Theorem 3.31 (H.~J. FINCK (Finc)): Let G be a connected regular graph 01 degree r BROOKS (see, for example, (Sac9)), with n vertices, where n - l' is even. Let Pi. be the multiplicity 01 the eigenval1le íl in the P-spectrum 01 G. For the ch1'matic n1lmber X(G) 01 the graph G the lollowing inequality x(G.) ;: r. + 1 (v = 1, ..., k), (3.37) holds: and so k k X(G) ;: r + min (Pr-n + 1, (n _~~ _ 1) + P2 + (p;iJ) - (n - 1') Pr-n' X(G) ;: ~ (1'. + 1) = ~ 1', + k = k(r + 1) - (k - 1) n. v=l v=! (3.38) This bound can be improved. In (3.37), equality holds only in the following four If n - r is odd, a more complex analysis is necessary. It seems that a problem cases of integer linear programming, similar to the one treated above (see Theorem 3.30), wil have to be solved. (a) rv = 0, (b) 1'. = 1,

(c) r" = 2 and Gv contains a component with an odd number of edges, 3.4. Some remai'ks on strongly regular graphs (d) rv;S 3 and Gv contains a complete graph with r" + 1 vertices as a component. If s is the number of graphs G, which satisfy one of these conditions, then Let x and y be any two distinct vertices of a graph and let LI(x, y) denote the number

k of vertices adjacent to both x and y. A regular graph G of positive degree r, not the X( G) ;: ~ Tv + s. complete graph, is called strongly reg1llar if there exist, non-negative integers e and 1 v=l such that L1(x, y) = e for each pair of adjacent vertices x, y and L1(x, y) = 1 for each We shall now derive an upper bound for s. In order to simplify the analysis we pair of (distinct) non-adjacent vertices x, y of G. shall assume that n - r is even. In this way graphs G. satisfying condition (b) The concept of a strongly regular graph was introduced by R. C. BOSE (Bas 1) are excluded since such graphs, on the one hand, must have n - l' + 1 vertices (1963), and at present there is already an extensive literature on this type of graph (this is an odd number) and, on the other hand, must have an even number of (see Sections 7.2 and 7.3). vertices. ( J From Theorem 1.9 (Section 1.8) we deduce immediately that a regular graph G The number of graphs Gv satisfying condition (a) is not greater than Po , of degree r / 0 - not the complete graph - is strongly regular if and only if there as mentioned earlier. n - r - 1 exist non-negative integers e and 1 such that the adjacency matrix A = (aij) of G A graph Gv satisfying (c) cannot be connected since its number of vertices n. satisfies the following relation: = n - r + 2 is even. This means that the characteristic polynomial of Gv contains a factor (íl - 2)2. Thus, according to Theorem 2.9, the characteristic polynomial A2 = (e - I) A + IJ + (1' - /) I. (3.40) of G contains a factor íl - 2 which- stems from Gv' So the number of such Gv is not greater than P2' Theorem 3.32 (S. S. SHRIKHANDE, BHAGWANDAS (ShBh)): A regular connected For each graph Gv which satisfies (d), a factor (íl + l)'v appears in the characteristic graph G 01 degree l' is st'1ngly regular il and only il it has exactly three distinct eigen- polynomial of G. Since rv ;S 3, the number of such graphs Gv is not greater than val1les íl(i) = r, íl(2, ),(3). ' Ii G is strongly regular, then

( p;il e = r + ),(2)íl(3) + ),(2) + íl(3) and 1 = l' + íl(2)íl(3). 104 3. Relations between spectral and structural properties of graphs 3.5, Eigenvectors 105

Proof. Let G be strongly regular. The eigenvalues of G are not all equal, for if they. Theorem 3.34: A multigraph is connected 1:f a.nd only if its index is a simple eigenvalile were, they would all be eqnal to zero - contradicting the hypothesis that G has an with a positive eigenvector. edge. Nor can the spectrum of G have exactly two distinct eigenvalues since then G Theorem 0.5 can also be translated into the language of graph theory: would have at least one edge and, according to Theorem 3.13, its diameter would be equal to 1 - contradicting the hypothesis that G is not the complete graph. I Theorem 3.35: If the ÙuJ'ex of a multigraph has multiplicity p, and if there Ù a. Since the relation (3.40) holds for G, the minimal polynomial of the adjacency positive eigenvector in the eigenspace corresponding to it, then G has exactly p comi)onents. matrix A of G has degree 3. Thus, G has exactly three distinct eigenvalues. Of particular interest are the eigenvectors of the line graph L(G) of a connected Let now G have exactly three distinct eigenvalues .í(1) = 1', .í(2), .í(3). ¡Then, ob- regular multigraph G of degree 1'.'1 Let G have 11 vertices and m edges. The relation viously, l' ? 0 and G is not the complete graph and according to Theorem 3.25, the connecting the spectra of G and L(G), namely relation PL(G)(/h) = (,u + 2)m-n PG(/h - l' + 2), (2.30) aA2 + bA + c1 =J (a =1 0) (3.41 ) has already been given in Theorem 2.15. holds, where ),(2) and ),(3) are the roots of the equation a.í2 + b.í + c = O. A com- Formula (2.30) establishes a (1, l)-correspondence between the sets of eigenvalues parison of the diagonal elements of the left and right side of (3.41) yields the equation .í =1 -1' of G and ,U =1 -2 of L(G): If ).=1 -r is a p-fold eigenvalue of Gtt, then ar + c = 1, or c = 1 - ar. /h = .í + l' - 2 =1 -2 is a p-fold eigenvalue of L(G), and if /h =1 -2 is a p-fold If the vertices i, j are adjacent, i.e. if aii = 1, we deduce from (3.41) that the eigenvalue of L(G), then .í = /h - l' + 2 =1 -1' is a p-fold eigenvalue of G. Therefore, number of walks of length two between i and j èquals 1 - b . If i, j are distinct we shall call (.í, p.) a pair of corresponding eigenvalues if ), =1 -r is an eigenvalue a of G, /h =1 -2 is an eigenvalue of L(G), and ), + l' = /h + 2. Denote the eigenspace belonging to the eigenvalue .í of G, or to the eigenvalue lt and non-adjacent, the corresponding number of such walks is l- . Hence, G is strongly a of L(G), by XV,) or Y(/h), respectively. Then the following theorem holds. regular. Comparing (3.41) and (3.40), we obtain e = l' + ),(2) + ),(3) + .í(2)),(3) and Theorem 3.36 (H. SACHS (Sac8)): Let G be a connected regular multigraph of degree l' f = l' + .í(2J.3). with 11 vertices an(l rn edges, let (.í, /h) be a pair of corresponding eigenvalues of G and This completes the proof. L(G), and let R denote the 11 X m vertex-edge incidence matrix of G. ThenRT maps the eigenspace X(.í) onto the eigenspace Y(/h), and R maps Y(/h) onto X(.í).

3.5. Eigenvectors Proof. As in Section 2.4, let A and B denote the adjacency matrices of G and L(G), respectively. In Chapter 1 we have seen that the eigenvectors of the adjacency matrix of a (multi-) 1. Let æ E X (.í), æ =1 0, and y = RTæ. Then, by virtue of (2.27) (Section 2.4), graph G, together with the eigenvalues, provide a useful tool in the investigation of Ry = RRTæ = (A + D) æ = (A + 1'1) æ = (J. + r) æ =1 0, the structure of G. In this section we shall go into a bit more detaiL. Sometimes valuable information about a (multi-)graph can be obtained from thus Y =1 o. Using (2.28) and again (2.27), we obtain its eigenvectors alone. Such a result is given by By = (RTR - 21) Y = RTRRTæ - 2y = RT(.í + 1') æ - 2y

Theorem 3.33: A multigraph G is regular if and only if its adjacency matrix has an = V, + r - 2) Y = /hY, eigenvector all of whose components are equal to 1. thus Y E Y(/h), i.e.: This theorem is a consequence of a well-known theorem of the theory of matrices æ E X(.í), æ =1 0 implies RTæ E Y(lt), RTæ =1 o. (see, e.g., (MaMi), p. 133). In (Bed), p. 131, the following result of r. H. WEI (Wei) is noted: t The line graph L(G) of a multigraph G with edges 1,2, ..., m is a multigraph with vertices Let Nk(i) be the number of walks of length k starting at vertex i of a connected graph G 1, 2, ..., m arid adjacency matrix B = (bik), where, for i =1 k, bik = 0 if the edges i, k of G have no vertex in common, bik = 1 if i, k are proper edges (i.e., not loops) having exactly one with vertices 1,2, ..., n. Let sk(i) = Nk(i). ,~Nk(j) . Then, for k -)- 00, the vector vertex in common, bik = 2 if i, k are proper edges having both their vertices in common or ()=1 n )-1 if one of them is a proper edge and the other one a loop haying a vertex in common, bik = 4 if i, k are both loops attached to the same vertex, and where bii = 0 if i is a proper edge, bii = 2 (Sk(l), sk(2), ..., sk(n))T tends towards the eigenvector of the index of G. if i is a loop. The question as to whether or not a given multigraph is connected can be decided "I Recall that Á = ~T is an eigenvalue (of multiplicity 1) of G if and only if G is bipartite by means of Theorem 0.4: combining Theorems 0.3 and 0.4, we obtain (see Theorem 3.4).

I 106 3. Relations between spectral and structural properties of graphs 3.5. Eigenvectors 107

2. In a similar way it can be proved that Theorem 3.38 (1\. DOOE (Do02), (Do08); for regular multigraphs see H. SACHS y E Y(f-), y =1 0 entails Ry E X(À), Ry =1 o. (Sac8)): Let 0 be any connected multigraph. Then y is an eigenvector ol L(O) belonging to the eigenvalue -2 il and only il 3. If Y E Y(f-), then there is a (unique) æ E XV,) such that RTæ = y, namely

1 I ~ y(u) = 0 lO1' each vertex v ol 0, which is equivalent to Ry = o. æ =-Ry. Ii'V f- + 2 The proof is contained in the proof of Theorem 6.11 (Section 6.3). 4. If æ E X(À), then there is a (unique) y E Y(f-) such that Ry = æ, namely the eigenvalue 1 Corollary to Theorem 3.38: Il y is an ei'genvector ol L(O) belonging to y=-RTæ. m À + r -2, then ~ Yi = 0, i.e.: the eigenspace Y( -2) is orthogonal to the vector (1, 1, ..., l)T. i=1 Theorem 3.36 is now proved. Hence, in line graphs the eigenvalue -2 never belongs to the main part ol the spectrum.

Remark. The mappings Rand RT considered in Theorem 3.36 can be given a more Note that, in a regular multigraph, each eigenvector which does not belong to intuitive form: the matrices R, A, B and the eigenvectors æ, y all refer to a fixed the index is orthogonal to (1, 1, ..., 1)T. labellng of the vertices and edges, respectively: for example, the component Xi of Recall that a regular spanning submultigraph (of degree s) of a regular multigraph an eigenvector æ of A corresponds to vertex Vi, so we may write x(v¡) instead of Xi, o is called a factor (s-factor) of O. or drop the subscript altogether and simply write x(v) for the component of æ that corresponds to vertex V and, similarly, y(u) for the component of y that corresponds We shall now establish, an interesting relation between the factors of 0 and the to edge u. eigenvectors of L(O). Now let a, b denote both vertices, or edges, or one of them a vertex and the other, A spanning submultigraph Of of a multigraph 0 can be represented by a vector one an edge, and let the symbol ~ mean that, for fixed b, the snmmation is to be c = (ci, C2, ..., Cm)T, with c¡ = 1 if the i-th edge of 0 belongs to Of, and c¡ = 0 a'b taken over the set of all CL which are adjacent to b or incident with b, respectively. otherwise; c is called the characte1'stic vector ol Of (with respect to 0). Then Let 0 be a connected regular multigraph of degree r. According to Theorem 2.15, y = RTæ is equivalent to the greatest eigenvalue of L(O) is 21' - 2 and the smallest eigenvalue is not smaller than -2. Let ¡yI, y2, ..., yPl be a maxinal set of linearly independent eigenvectors y(u) = ~ x(v) for each edge u, belonging to the eigenvalues of L(O) which are greater than -2 and smaller than V'II 21' - 2. It is easy to see that p = n - 2 if 0 is bipartite and p = n - 1 in the oppo- æ = Ry is equivalent to site case; if n / 2, then p / O. Denote the m X p matrix with columns x(v) = ~ y(u) for each vertex v. yI, y2, . .., yP by JU. 'U''V The next theorem provides a nieans for the investigation of the existence of an Next the eigenvectors æ of 0 and y of L(O) belonging to -1', -2, respectively, s-factor in 0, providing ltI is known. shall be investigated. Theorem 3.39 (H. SACHS (Sac8), (Sac14)): Let 0 be a connected regular multigraph Theorem 3.37: Let 0 be any connected multigraph. Then 0 is bipartite il and only ol degree r with n vertices and m edges. A vector z with m components is the characteristic il the system ol equations vector ol an s-lactor ol 0 il and only il it satisfies the following conditions:

~ x(v) = 0 lO1' each edge u ol 0, (3.42) 1 V.'U 10 _ sn components of z are equal to 1, (ind the other components are equal to zero; equivalent to RTæ = 0, 2 2° JUTz = O. has a (unique) non-trivial solution. Il, in particular, 0 is regular of degree l' and bipartite, the'l the solution ol (3.42) Proof. 1. Let z be the characteristic vector of an s-factor Os of O. Then 1° holds equals the eigenvector belonging to )'11 = -1'. trivially. In order to deduce condition 2°, consider the vector y = rz - syO, where The simple proof may be left to the reader. yO is a vector all of whose components are equal to 1 (note that yO is the eigenvector 108 3. Relations between spectral and structural properties of graphs 3.5, Eigenvectors 109 of the index 21' - 2 of L(G)). Then Yi = l' - s if the j-th edge belongs to G" and Hence, the numbèr of linearly independent characteristic vectors of regular factors is Yi = -s otherwise. Further, not greater than m - p. So, if we call a set of regular factors independent (dependent) if the corresponding characteristic vectors are linearly independent (dependent), ~ y(u) = s(r - s) + (1' - s) (-s) = 0 n"v we have for each vertex v of G (or, briefly, Ry = 0). According to Theorem 3.38 this means Theorem 3.40 (H. SACHS (Sac 8), (Sac14)): The number of independent regula,r that y is an eigenvector belonging to the eigenvalue -2 of L(G), so y is orthogonal factors of a connected regulaT miÛtigraph G of degTee T with n vertices and m edges is not greater than to y1, y2, ..., yp. Since yO has the same property, z = ~ (y + syO) is also orthogonal to each of y1, y2, ..., yp. Hence, MTZ = o. l' . if G is bipartite, 2. Let now 1° and 2° hold for a vector z with m components. Let y = 1'Z - syo. Then m-p= 1 -m-n+1 otheTwise.t \,fyO, y) = l' .1 -, 2sn - sm1 =2 l' . - sn 1-2 s . - rn = 0 i ~m-n+22 Moreover, H. SACHS (Sac 8) proved by direct construction that this bound is and for each vector yi (i = 1,2, ..., p) attained for each n, T with nT 0 (mod 2) by some multigraph, for n even in both cases, and £01' n odd - naturally - only by non-bipartite multigraphs. (yi, y) = r(yi, z) - S(yi, yO) = O. Several further results concerning eigenvectors of multigraphs can be found in Hence, y is orthogonal to yO, yl, ..., yP and therefore y is an eigenvector belonging Sections 5.1, 5.2, 6.3. to the eigenvalue -2 of L(G). According to Theorem 3.38, ~ y(u) = 0 for each u'v Remark (H. S.). Many of the results stated above for regular multigraphs can be vertex v of G. Since the components of yare r - s or -s, it follows from the last generalized to arbitrary multigraphs if, instead of the ordinary spectrum (= P-spec- equation that for exactly s of the edges which are incident with v the equation trum), the Q-spectrum and the corresponding eigenvectors (= Q-eigenvectors) are y(u) = r - s hplds and that y(u) = -s for the remaining r - s edges. Those edges utilized. (Recall that x is called a Q-eigenvector belonging to the Q-eigenvalue ). u for which y(u) = l' - s (these are precisely the edges for which z(u) = 1) form an if x =! 0 and x and ), satisfy Ax = ÀDx.) s- factor of G. This completes the proof of the theorem. The following propositions can easily be proved. Let G be a connected multigraph. Then Corollary to Theorem 3.39: If a vector z with m components, q of which a1'e equal to 1 and the other m - q of which are equal to 0, satisfies the conditionMTz = 0, then 1 ° all Q-eigenvalues Ài are Teal and satisfy - 1 ~ Ài :: 1; 20 the maximal Q-eigenvalue À1 (= "Q-index") is a simple eigenvalue equal to 1; 2q 0 (mod n) and z is the characteristic vector of ans-factor with s = 2q. n 30 the Q-eigenvector belonging to )'1 is (1, 1, ..., 1) T ; The proof of the Corollary is left to the reader. 40 any two Q-eigenvectoTs x, Xl belonging to different Q-eigenvalues (/.re orthogonal in Let"l be a subset of the set q¡ of edges of G. Then"l induces a regular factor if the following sense: x TDxl = 0; and only if 5° the following statements are equivalent: ~ yi(U) = 0 (i= 1,2,...,p). (i) G is bipartite, UE"f (ii) the Q-spectTum of G is symmetric with Tespect to the zero point of the Teal axis, Let SP be the vector space generated by y1, y2, ..., yp. Clearly, each vector of SP is a (iii) -1 is a (necessaTily simple) Q-eigenvalue; solution of the following system of homogeneous linear equations 60 if G is bipartite, then the Q-eigenvector x belonging to Àn = - 1 satisfies Xi + Xk = 0 for each pair i, k of adjacent vertices. L Y( u) = 0, UE"f In the sequel, suppose that G is an arbitrary connected multigraph with n ~ 1 vertices and'm ~ 1 edges. where"l runs through all subsets of o¡ which induce a regular factor. Therefore the rank of the coefficient matrix of this system is not greater than m - p. The l' Note that-. rn - n + 1 = m - n + 1 is the cyclomatic number of G. rows of this matrix are just the characteristic vectors of the regular factors of G. 2 \ 110 3. Relations between speGtral and structural properties of graphs 3.5. Eigenvectors 111

Now introduce two "modified incidence matrices" and Proposition 5° (iii) may now be restated as follows:

a is bipartite if and only if 2;' = O. R* = D-IR, S* = -. RT. (3.43) 2 By (3.47) a simple multiplicity preserving (1, I)-correspondence between the sets of Both of them are stochastic with respect to their rows, and so are the "modified all non-zero (i.e., positive) eigenvalues 2* of A* and eigenvalues ,u* of B* is given, adjacency matrices" namely 2* = ,u*.

A* = R*S*, B* = S*R*.t (3.44) If 2 is a Q-eigenvalue of a, let X(2) be the corresponding eigenspace; then, clearly, Clearly, 1 1 1 withthe eigenspace X*(J.*) X(J,): of the eigenvalue ),* =2 2 + 1 of the matrix A* is identical A* = - D-IRRT = - D-I(A + D) = - (D-IA + I), (3.45) 2 2 2 X*(J.*) = X(2).

B* = -. RTD-1R = B*T. (3.46) Denote the eigenspace of the eigenvalue ,u* of the matrix B* by Y*(,u*). Then the 2 analogue of Theorem 3.36 holds: Define the characteristic polynomials H Theorem 3.36': Let 2 * = ,u* ? 0 be corresponding eigenv(ilues of A *, B*, respectively,

fG(t) = It I - A*I, gG(t) = ItI - B*I and let J, = 22* - 1 be the corresponding Q-eigenvalue ? - 1. Then S* maps the with corresponding spectra eigenspace X(J,) = X*(J.*) onto the eigenspace Y*(,u*), and R* m(ips Y*(,u*) onto X*(2*) = X(J,).

P.i, 2~, ..., 2;,)¡, (,ui, ,u~, ..., ,uii.)g' Theorem 3.38 has the following analogue: By (3.44) Theorem 3.38': y* is an eigenvector of B* belonging to the eigenvalue ,u* = 0 if and tmfG(t) = t"gG(t), (3.47) only if R*y* = o. and because of (3.46), all eigenvalues ).1', ,uj are real numbers. Now define a generalized factor of an arbitrary multigraph a = (!!, %') as follows: By virtue of (3.45), D-IA = 2A* - I, so Let the valencies d(v) of the ve~.tices v E !! have the greatest common divisor 0, and let 0 .c a ~ o. A spanning submultigraph a' = (!!, %") of a is called a a-factor if QG(t) = It I - D-IAI-jtI - 2A* + II = 2" It ~ 1 I -A*I = 2"fGC ~ 1). for every vertex v E !! the valencies d(v) with respect to at and d(v) with respect to a have the same ratio a: ö, i.e., if Consequently, if (21,22, ..., J,,,JQ is the Q-spectrum of a, a d(v) = - d(v) for every vertex v E !!. o 2:l'i-- _ 2i +2 1 ' 2i = 22: - 1 (i = 1,2, ..., n). (3.48) A non-trivial a-factor can, of course, exist only if 0 ? 1. In connection with (3.47) and (3.48), Proposition 10 yields Let ¡æ1, æ2, ..., æPj be a maximal set of linearly independent Q-eigenvectors belonging to the Q-eigenvalues of a which are greater than - 1 and smaller than 1. o ~ 21 ~ 1 (i = 1,2, ..., n), (3.49) Clearly, p = n - 2 if a is. bipartite and p = n - 1 otherwise. Denote the n X p o ~,uj ~ 1 (j = 1,2, ..., m), matrix with columns æi by E ançl put M* = 2S*E = RTE; then, according to Theorem 3.36', the p columns y*i of M* constitute a maximal set oflinearly independ- t Note that the definitions ofR*, S*,A*,B* may be e:itended to hypergraphswithR* = D;lR, ent eigenvectors belonging to the positive eigenvalues ,u* of B* which are smaller S* = D¡lRT, where Dv is the "valency matrix of the set of vertices" and DB is the "valency than 1. Now the following theorem which is a generalization of Theorem 3.39 can matrix of the set of (hyper- )edges". These definitions lay a basis for a theory of a modified be proved in a way analogous to the proof of Theorem 3.39. "Q-spectrum of a hypergraph" and at the same time explain the apparent asymmetry in the definitions of R*, S* given above (formula (3.43)). (See also Section 1.9, nos. 11, 13.) tt When dealing with hypergraphs, F. RUNGE (Rung) has used polynomials similar Theorem 3.39': Let a be a connected multigraph with m ~ 1 edges. A vector z with to fG(t) and gG(t); see Section 1.9, nos. 11, 13. m components is the characteristic vector of a a-factor of a if and only if it satisfies the 112 3. Relations between spectral and structural properties of graphs 3.6. Miscellaneous results and problems 113 following conditions: 7. Let A be'the adjacency matrix of a graph G with n vertices, let f c: 11,2, ..., nj and let a Af be defined as in Section 1.5 (p.37). Then the number of Hamiltonian circuits of Gis, 1° - In components of z aTe equal to 1, and the otheT components of z aTe equ,Û to 0; given by o 2° J1tH Z = o. .. ~ (_1)8 L tr A:ø. 2n 8~0 Ifi~8 (L. lV1. LIHTENBAUlIi (JIl1x4j, (JIux5)) An analogue to the Corollary to Theorem 3.39 is also valid. 8. Let P_, Po' p+ denote the number of eigenvalues of a graph G, which are smaller than, equal The relations between the (generalized) factors of a (multi-)graph and the to, or greater than zero, respectively. Then the chromatic number X(G) of G satisfies eigenvectors of its line graph become particularly evident when extended to hyper- graphs (see also footnote on p. 110). X(G)~Po + mm .n (p+, .p-) , where n is the number of vertices of G. This inequality is sharp; equality holds, for example for complete graphs. 3.6. Miscellaneous results and problemS (D. M. CVETKOVIC (Cvell))

1. Let e and e be the indices of the graphs G and G. Let G have n vertices. By use of Theorem 9. Let G be a regular graph of degree r = n - 3 with n (n ~ 3) vertices. Put, 3.8 and relation (7.29), the following inequalities are easily obtained: (_1)n (), + n - 2)-i (À - 2) PG(-Â - 1) = À"+ ai).n-i +... + an n - 1 ~ e + e ~ Y2(n - 1). = (À - 2)"" (À + 2)"'-' qi(Â), The left-hand inequality is actually an equality if and only if G is regular. where qi(2) =1 0 and qi( -2) =1 O. Then the chromatic number X(G) of G is given by 1 (E. NOSAL (Nos1); A. T. AMIN, S. L. HAKIMI (AmHa)) X(G) = - (n + m2 - m_2 + aa)' 2 2. Prove that Ài ~ Ydm.x, where Ài = e is the index and dmax is the maximal valency of a graph. (H,-J. FINCK (Fine))

(E. NOSAL (Nos1); L. LovÁsz, J. PELIKÁN (LoPe)) 10. Show that the chromatic number X of a graph G is determined by the spectrum of G if the index e of G is smaller than 3. If G is connected, the same statement holds also for e = 3. 3. Let G be a regular graph of degree r with n vertices. Show that for the number t(G) of span- (D. lV1. CVETKOVIC (Cve9J) ning trees of G the following formula holds: 11. For a given k let a, b, c, d, e denote the numbers of eigenvalues  in the spectrum of a graph t(G) (_1)n=-P(j(-T-1). G, which, respectively, satisfy the following relations: À ~ -k + 1, ), = -k + 1, -k + 1 n2 ~ À ~ 1, À = 1, Â:? 1. Further, let 8 be the smallest of the natural numbers k (k :? 1) for 1 1 which the inequality 4. If G is a connected graph, neither a tree nor a circuit, then e :? T"2 + T -"2, where e is the min (b + c + k(d + e), k(a + b) + (k - 1) (c + d)) ~ n index of G and T = .. (Võ + 1). holds. Then X(G) ~ 8. 2 (D. lV1. CVETKOVIC (Cve 12)) (A. J. HOFFMAN (Hof 13)) 12. Let X(G) be the chromatic number of the complement G of a graph G. If G is not a complete 5. Show that the star has the largest index among all trees with n vertices. graph, then 6. Each closed walk in a graph G can be represented as a sequence of vertices through which X_ (G)- ¿ 1+À2n + À2 - Âi' it passes, for example, Xi' X2, ..., Xn, Xi' The walks X¡, X2' ..., Xk-i' Xh Xl and X2' xa' ..., Xk, X¡, x2 are different because one starts from. xi and the other from X2 but are considered as cyclically where n is the number of vertices of G and Ài, )'2 are the first two greatest eigenvalues of G. equivalent: Two closed walks are called cyclically equivalent if one is obtained from the other (A. J. HOFFMAN (Hof 16)) by rotating an initial segment to the end of the walk. Let Ck(G) be the number of cyclic equi- valence classes of closed walks of length k in G. Then 13. Let a(G) be the smallest number ,of subsets into which the vertex set of G can be parti- tioned such that the subgraph induced by anyone of thè subsets is either a complete graph 1 (k) n d or a totally disconnected graph. Let k be a positive integer and let G have eigenvalues Ài, ..., ).". Ck(G) =-Lk dlk ø d- i=iL À¡, Then there exist functions d k and éi k such that a(G) ~ dk(À2 - Ân-k+i), a(G) ~ PJk(J'k - Ân). where Ø(k) denotes the Euler phi-function and Ài, ..., Àn are the eigenvalues of G. À similar (... J. HOFFMAN, L. HOWES (HoHo)) formula was obtained also for the number of dihedral equivalence classes of closed walks in a graph. 14. If 1/"(G) is the vertex set of a graph G and if .Y c: "f(G), then G!7 denotes the subgraph (F. HARARY, A. J. SCHWENK (HaS 1)) of G induced by the set of ve,rtices of G each of which is adjacent to all vertices in .Y. Let

8 Cvetkovic/Doob/Sachs 3.6. Miscellaneous results and problems 115 114 3. Relations between spectral and structural properties of graphs

21. The eigenvalues and eigenvectors of C = D - A (D is the valency matrix of a graph) of Gandlet k(H) denote the cliquomatic numberÎ of a graph H. ,1*(G) be the smallest eigenvalue were used by K. M. HALL (Ha, K) in a problem of minimizing the total length of the edges Then of a graph which is to be imbedded into a plane. 10 there exists a function f such that if x E 1/'(G),

k(G(x1) ~ Ip.*(G)); 22. Define a relationship"" on the vertex set of a graph G thus: x .. y if for every z =f x, y the vertex z is adjacent to both or to none of vertices x, y, Let e(G) denote the number of 20 there exists a function g such that if x E "f(G), equivalence classes so defined. e(G) is not determined by the spectrum of G (see Fig. 6.1). But A. J. HOFFMAN (Hof17) has proved that e(G) is bounded from above and below by some Ili i i is not adjacent to x, I"(G(x,iJ)I ? g(,1*(G))lI ~ g(,1*(G)). (A. J. HOFFMAN (Hof 15)) functions of the number of eigenvalues of G not contained in the interval (~ (V'S - 1),1).

15. Prove that the Seidëi (-1, 1, OJ-spectrum (see Section 1.2) of a self-complementary graph is symmetric with respect to the zero point. 23. Let G be a regular graph of degree l' with n vertices. Let Gi be an induced sub graph of G having ni vertices and average vertex degree 1'1" Then 16. Let R be the vertex-edge incidence matrix of a connected multigraph G having n vertices. Then ni(i' - ,1,,) , , ~, .: ni(i' - ,12) + ' if G is bipartite, Î "'" = i i = "'2' J n - 1 n n rk R This inequality was derived in (BuCS) by the use of Theorem 0.11. Specifying I'i = 0 and t n otherwise. (H. SACIrS (Sac8); C. VAN NUFFELEN (Nuf1)) Ti = ni - 1 we get the following inequalities for the-cardinalities Ix(G) and K(G) of an internal stable set and of a complete subgraph of a regular graph G, respectively: 17. Let the edges Ui, U2,... U2k form a closed walk (of even length) in a regular multigraph G and let y be an eigenvector of L(G) not belonging to the eigenvalue -2. Then Ix(G) ~ -nÀ" K(G) ~ (,12 + 1) n 'i" - ,1,,' n-r+,12' 2k 2: (_l)i Y(u¡) = O. where ,12 and ,1,, are the second largest and the least eigenvalue of G. The first bound was found ;=0 (H. SACHS (Sac 8)) by A. J. HOFF~IAN (unpublished). Together with Ix(G) X(G) ~ n it gives the bound from Theorem 3.18 in the case of regular graphs. Similarly, the second inequality gives the bound 18. Let G be a graph with eigenvalues ,1i, ..., ,1n- Let c(G), #(G), and o(G) be the quantities for the cliquomatic number (see Section 3.6, no. 12). Specifying Ti in some other ways we defined as in Theorem 3.21. Then can get bounds for some more characteristics of a graph. The inequality for Ix(G) in the case c(G) = 1 if and only if ,12 ~ 0, of strongly regular graphs was noted in (Del1). The inequality can be extended to non-regular graphs (W. HAEMERS (Haem)). #(G) = 1 if and only if ,1,,-i ~ 0, o(G) = 1 if and only if ,12 ~ 0 and ,1,, = -1. 24. Further bounds for K(G), defined in Theorem 3.15, can be obtained by the same technique (A. J. HOFFMAN (Hof11)) using the Seidel adjacency matrix instead of the (0, l)-adjacency matrix. For example, J. M. GOETHALS and J. J. SEIDEL (GoS4) found the inequality K(G) ~ min (1 - 122' fli' fl2), where 19. Let G be a graph and let cm(G) be the smallest integer k such that (3.22) holds and each G is a graph whose Seidel adjacency matrix has only two distinct eigenvalues 12i. 122 (l2i ? 122) G; is a complete s-partite graph with s ~ m. Then with the multiplicities fli' fl2' respectively. Generalize this result for arbitrary graphs and, cm(G) ~~, in the case of regular graphs, express it in terms of the eigenvalues of the (0, l)-adjacency m - 1 matrix. where p_ is the number of negative eigenvalues of G. 25. If D; (i = 3,4,5) are the numbers of circuits of length i in a regular graph of degree 1', (D. T. MALBAŠKI, private communication) and if ai (i = 0, 1, ..., n) are the coefficients of the corresponding characteristic polynomial, then 20. A k-partition of a graph is a division of its vertices into k disjoint subsets containing 1 mi, m2, ..., mk vertices, respectively, where mi ~ m2 ~ ... ~ mk' D3 = - - a3, Let G be a graph with adjacency matrix A, let U be any diagonal matrix such that the 2 sum of all the elements of A + U is zero, and let fli, fl2' ..., flk (fli ~ fl2 ~ ... ~ flk) be the 1 2 D4 = - (a2 + 2m2 - a2 - 2a4) , largest k eigenvalues of A + U. Then, if any k-partition a of G is given, the number Eo of 4 edges of G whose two vertices belong to different subsets of a satisfies D5 = -1 (a3a2 + 3ra3 - 3a3 - a5). Eo ~ -1 2:k (-mxflx)' 2 2 x~i 26. If )'i' ..., ,1,, are the eigenvalues of a regular graph G,then the number D4 of circuits of The right-hand sum is a concave function of U. length 4 in G is given by . (W. E. DONATH, A. J. HOFFMAN (DoHo), cf. also (Fie3)) D4 =.. (~ ,1t - n¡(2,1i - 1)) . -r The cliquomatic number of a graph G is the chromatic number of the complement G of G. 8 ;=1 8*