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University Microfilms International 300 North Zeeb Road Ann Arbor, Michigan 48106 USA St. John's Road, Tyler's Green High Wycombe, Bucks, England HP10 8HR 77-24,677 NEMZER, Daniel Edward, 1947- CHARACTERIZATION OF LINE GRAPHS. The Ohio State University, Ph.D., 1977 Mathematics
Xerox University Microfilms, Ann Arbor, Michigan 4Bioe CHARACTERIZATION OF LINE GRAPHS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By Daniel Edward Nemzer, B.S., M.S.
*****
The Ohio State University
19 77
Reading Committee* Approved By Professor Dijen K. Ray-Chaudhuri t to s'* nf Professor Thomas A. Dowling • 7 — ( k.. Professor G. Neil Robertson Adviser Professor Richard M. Wilson Department of Mathematics ACKNOWLEDGMENTS
I am deeply grateful to my adviser, Prof. Dijen K. Ray-Chaudhuri. He guided me in the research reported in
this dissertation. The members of the reading committee made many helpful suggestions.
Prof. Arnold E. Ross (at The Ohio State University), and Prof. Frederick B. Thompson (at the California Institute of Technology), are among the teachers who have inspired my studies. Mrs. E. L. Karlquist, of the OSU Mathematics Library, helped me to locate many useful journal articles. This dissertation is dedicated to my mother, Mrs. Louis Nemzer, and to the memory of my father, the late Prof. Louis. Nemzer. The loving guidance of my parents has been a blessing. Finally, I thank my wife, Elaine. She has been a constant source of encouragement and understanding. VITA July 8, 19^7 • • Born— Washington, D.C. 1969 ...... B.S. with Honor, California Institute of Technology, Pasadena, California 1969-1971 • • • University Fellow, The Ohio State University, Columbus, Ohio 1970 • . . . M.Si,' The Ohio State University, Columbus, Ohio 1970-1971 • • • Graduate Teaching Associate, Department of Mathematics, The Ohio State University 1971-1973 • • • Mathematician and Computer Analyst, U.S. Public Health Service, Rockville, Maryland
1973-197^ • • • Graduate Teaching Associate, Department of Mathematics, The Ohio State University 197^f ...... Graduate Research Associate, Department of Mathematics, The Ohio State University
1973-1975 • • • University Fellow, The Ohio State University, Columbus, Ohio
FIELDS OF STUDY Major Field* Mathematics Studies in Combinatorics and Graph Theory* Professors D. K. Ray-Chaudhuri, Thomas A* Dowling, G. Neil Robertson, and Richard M. Wilson Studies in Algebra* Professor Harold D. Brown Studies in Number Theory* Professor Arnold E. Ross
iii TABLE OP CONTENTS
Page ACKNOWLEDGMENTS ...... ii VITA ...... iii LIST OP T A B L E S ...... vi LIST OF FIGURES ...... vii INTRODUCTION ...... 1 Chapter I. DEFINITIONS ...... 11 1• Graphs 2. The line graph of a graph 3* The line graph of a tree 4. The eigenvalues of a graph 5. Literature review II. THE LINE GRAPH OF A BALANCED INCOMPLETE BLOCK DESIGN ...... 30 1. Introduction 2. The line graph of a semiregular bigraph 3. Nonedge degree at most 1 4. The parameters of a BIBD 5* No subgraph K^-Kg 6 . The line graph of a BIBD III. GENERALIZED LINE G R A P H S ...... 49 1. Introduction 2. Cocktail-party graphs 3* Generalized line graphs 4. Permitted graphs 5• Forbidden graphs IV. GRAPHS WITH LEAST EIGENVALUE £ - 2 ...... 67
1. Introduction 2. The edge-signed graph of inner products 3. The base graph of a set of vectors 4. Root systems 5* The base graph of a graph V. CONSTRUCTION OF EXCEPTIONAL GRAPHS ...... 100 1. Introduction 2. Construction from a base graph 3. Exceptional trees 4. Graphs represented in E^ 5• Exceptional graphs VI. THE LINE MULTIGRAPH OF A M U L T I G R A P H ...... 125 1. Introduction 2. Multigraphs 3. The line raultigraph 4. Geometric characterizations 5. Multigraphs with least eigenvalue ^ -2 APPENDICES A. THE 143 CONNECTED GRAPHS ON 1 TO 6 VERTICES . 144 B. DIAGRAMS OF REGULAR EXCEPTIONAL GRAPHS .... 149 C. DIAGRAMS OF NON-REGULAR EXCEPTIONAL GRAPHS . . 156 BIBLIOGRAPHY ...... 161
v LIST OF TABLES Chapter I Page
Table 5 • 1 ■ Characterizations of graphs 26
Chapter II Table 1.1. Geometric characterizations of the line graph of a BIBD 31 Table 4.1. The line graph of a nontrivial, nonsymmetric BIBD 42
Chapter IV Table 3.1. The base graph of a set S of vectors 78 Table 5-1. The base graph of a graph H 93
Chapter V Table 4.1. The 42 patterns of X, Y, W in 113 Table 4.2. An upper bound on the number of exceptional graphs in E^ 119
Appendix A Table A.I. The number of graphs on 1 to 6 vertices 144 Table A.2. Eigenvalues of the 143 connected graphs on 1 to 6 vertices 146
Appendix B Table B .1. The 187 regular connected exceptional graphs 150 LIST OF FIGURES
Chapter I Page Fig. 1.1. The graphs Kj, K^, K^-Kg, and ^ 12 Fig. 2.1. Ten connected graphs and their line graphs 15 Fig. 2.2. Odd and even triangles 16 Fig. 2.3* The nine graphs forbidden in a line graph 18 Fig. 3.1. An even triangle in H = L(G) 22 Fig. if-.l. Two isospectral graphs 2k Fig. 5•1• The two Kuratowski graphs, and ^ 29
Chapter II Fig. 3.1. Two graphs which are not line graphs 39 Fig. 5-1. The graph K k-K ^3 Fig. 5.2. A four-cycle in the line graph of a BIBD k5
Chapter III
Fig. 3*1* A graph and a generalized line graph 52 Fig. k.l. j in two generalized line graphs 55 Fig• k .2. Ktj-Kg in four generalized line graphs 56 Fig. ^.3. The graph 57 Fig. k,k* The graph 59 Fig. 5•1• The nine graphs forbidden in a line graph 60 Fig. 5.2. K^-Kg in a generalized line graph 61 Fig. 5*3* K^-Kg not in a generalized line graph 61 Fig. 5-k. Gg not in a generalized line graph 62 Fig. 5.5. G^ not in a generalized line graph 63 Fig. 5.6. G- not in a generalized line graph 64 Fig. 5.7. Gg not in a generalized line graph 65 Fig. 5*8. Gg not in a generalized line graph 68
Chapter IV
Fig. 2 .1. Three adjacent vectors in a star or cone 71 Fig. 2 .2 . The edge-signed graph of a,b,c,x^ 72 Fig. 2.3. The edge-signed graph of a,b,c,y2 72 Fig. 2.4. The edge-signed graph of a fbtc(z^ 73 Fig. 2.5. The edge-signed graph of a,b,c,x^,3/ri,zi 74 Fig. 2 .6 . Type 1 connection, with (x^»x^) - 1 75 Fig. 2.7. Type 0 connection, with (x^,x2) = 0 76 Fig. 3.1. The line graph L(K^ 80 Fig. 3.2. The line graph Xj(K^5 81 Fig. 3 0 . Segre coordinates for the Schldfli graph 82 Fig. 3.4. Inner products of Cameron et al. (1978) 83 Fig. 4.1. The graphs Q, y and K^-LI 90 Fig. 5.1. H does not contain K2 ^ or K^-LI 91 Fig. 5.2. Type 1 connection, with (xi»xj) = 1 94 Fig. 5.3. Type 0 connection, with (x^.x.) - 0 94 Fig. 5.4. H does not contain Q or K^-1I 97
Chapter V
Fig. 3.1. Cycles in H are not permitted 105 Fig. 3.2. The six base graphs of exceptional trees 106 Fig. 3*3* The six T vertices in Eg 107 Fig. 3.4. The two exceptional graphs in Eg 107 Fig. 3-5- Signs for T in , and the graph H 109
• • a V l l l Pig. 3.6. Signs for T in B^, and the graph H 110 Fig. 4.1. Substitutions for Pattern 1 118 Pig. 5.1. The twenty smallest exceptional graphs 124
Chapter VI Fig. 3 .1 . A 4-bond in M, and a 4-cluster in L(M) 130 Fig. 3-2. The three forbidden multigraphs 130 Fig. 3 0 . Five underlying graphs G, G* 133 Fig. 3.4. Three cluster-reduction graphs H, H' 133 Fig. 3 .5. Multigraphs and line multigraphs 134 Fig. 4.1. The nine graphs forbidden in a line graph 135
Appendix B
Fig. B. 1. Some regular exceptional graphs 151 Fig. B.2. BCS #-'5» "the Petersen graph 152 Fig. B.3. BCS # 185» the total graph T(C^) 152 Fig. B.4. BCS # 186 153 Fig. B.5. BCS # I871 the Clebsch graph 153 Fig. B.6. BCS #. 1 154 Fig. B.?. BCS # 9» the Hofftnan and Ray-Chaudhuri graph 154 Fig. B.8 . BCS # 69, the Shrikhande graph 155 Fig. B«9* BCS # 184, the Schiafli graph 155
Appendix C
Fig. C .1. Exceptional graph 157 Fig. C.2. Exceptional graph 157 Fig. C .3. Exceptional graph 157 Fig. C.4. Exceptional graph 157
ix Fig. C.5. Exceptional graph 158 Fig. C.6. Exceptional graph 158 Fig. C.?. Exceptional graph 158 Fig. C.8. Exceptional graph 159 Fig. C.9. Exceptional graph 159 Fig. C.10. Exceptional graph 159 Fig* C.ll. Exceptional graph 160 Fig. G.12. Exceptional graph 160
X INTRODUCTION
The characterization of graphs has heen a lively topic in Graph Theory for the past twenty years. This dissertation proves some new characterizations of line graphs, by their geometric and spectral (eigenvalue) properties• A graph G is a finite nonempty set of elements called vertices. together with a set of edges, where an edge is an unordered pair of distinct vertices. By this definition, a graph has no multiple edges, no loops, and no directed edges. A graph expresses the relationships within a set of elements. For this reason, graphs arise naturally in applications to Chemistry, Physics, Electrical Engineer ing, Linguistics, Sociology, and Information Theory. The line graph of a graph G, is a graph L(G), whose vertices correspond to the edges of G. Two vertices of L(G) are adjacent if they correspond to two edges of G that have one common vertex. The line graph provides a new perspective on a graph. For example, the map of a state contains a graph of the cities (vertices), joined "by highway segments (edges). The line graph would he a schematic chart of the highway segments (vertices of the line graph), with a line (edge of the line graph) joining two highway segments if they meet in a city. The eigenvalues of a graph are the eigenvalues (p. such that Ay - |uv for v / 0 ) of the adjacency matrix of the graph. In Graph Theory, the eigenvalues are numerical invariants of the graph. Unfortunately, the eigenvalues do not uniquely identify the graph. ■ Collatz and Sinogowitz (1957)* showed that there are non-isomorphic graphs with the same eigenvalues. In Chemistry, a molecule can he drawn as a graph, with the atoms as vertices, and bonds as edges. The first investigation of the eigenvalues of a graph was by two chemists, Coulson and Rushbrooke (19^0). They studied the Htickel approximation to Schrfldinger•s equation. For a class of hydrocarbons known as conjugated polyens, if 0 is an eigenvalue of the graph of a molecule, then the molecule cannot have total electron spin equal to zero. This implies that the molecule is chemically unstable (Cvetkovic and Gutman 1972).
* The citation "Author (Year)" refers to a journal article by that Author, published in that Year. All articles cited in the text are listed in the Bibliography, alphabetically by Author and chronologically by Year. Outline This dissertation has six chapters. Chapter I defines line graphs. We characterize the line graph of a tree, and the line graph of a graph with one cycle. There is a literature review of geometric and spectral characterizations of graphs. In Chapter II, we prove geometric characterizations of two families of graphs. The first is the line graph of an (r,1c)-semiregular bipartite graph. The second is the line graph of the bipartite graph of a balanced incomplete block design (BIBD). This generalizes a result of Rao and Rao (1969)* It also completes a series of geometric characterizations of line graphs of BIBDs, begun by Shrikhande (1959 b).* Generalized line graphs (Hoffman 1970) are introduced in Chapter III. This is an important class of graphs, with least eigenvalue greater than or equal to -2. There are nine graphs which cannot be induced subgraphs of a line graph. We prove that two of these can be induced subgraphs of a generalized line graph, and seven cannot. Chapter IV presents the theory of graphs with least eigenvalue greater than or equal to -2. We apply the results of Cameron, Goethals, Seidel, and Shult (19?6) to
* The citation (1959 b) indicates the second (b) article by that author in that year. vectors, instead of lines, in Euclidean space. A graph with least eigenvalue £ -2 is the graph of inner products of a subset of vectors in a Lie algebra root system Dn or Eg. We define a base graph B(H*ab)» which is used to characterize graphs with least eigenvalue ^ -2.
Exceptional graphs are constructed in Chapter V. A graph is exceptional if it has least eigenvalue greater than or equal to -2, but it is not a generalized line graph. In the notation of the previous chapter, the base graph B(Hiab) of an exceptional graph is an induced subgraph of the Schldfli graph. The base graph of an exceptional graph represented in the root system Eg, is an induced subgraph of the line graph L(K~ «). j t J Chapter VI is about multigraphs. A multigraph is a finite nonempty set of vertices and a family of edges. Thus two vertices in a multigraph may be joined by more than one edge. Loops are not allowed. The vertices of the line multigraph L(M) of a multigraph M, correspond to the edges of M. Two vertices of L(M) are 1-adjacent or 2-adjacent, when the corresponding edges of M have 1 or 2 ends in common, respectively. We prove analogs, for line multigraphs, of the geometric characterizations of line graphs. Finally, we characterize multigraphs with least eigenvalue greater than or equal to -2. Statement of Results We have collected "below the statements of the main theorems proved in each of the six chapters of the text. Definitions are given here for some of the terms that occur in the theorems. All technical terms are defined in the text. Within Chapter I, the first theorem of Section 3 is called "Theorem 3.1." Outside of the Chapter, the same theorem is referred to as "Theorem I.3 .1 ."
A tree is a connected graph with no cycles.
Theorem 1.3.1. Let H be a connected graph. Then H is the line graph of a tree, if and only if none of the following graphs is an induced subgraph of H* (1) An n-cycle Cn for n > 4 ,
(2) k i (3 • °r (3 ) k4 -k2 .
Theorem 1.3.2. Let H = L(G) be a connected line graph, H not isomorphic to or to K^-Kg. Then G has exactly one cycle, if and only if either* (1 ) H has exactly one induced subgraph Cn for n ^ k , and H has no even triangle, or
(2) H has no induced subgraph Cn for n ^ 4 , and H has exactly one even triangle. A 'bipartite graph (or bigraph) is a graph whose vertex set can be partitioned into two subsets, such that no two vertices in the same subset are adjacent. A bigraph is ( r. k) - semiregular if* each vertex in one subset has degree r, and each vertex in the other subset has degree k. The degree of an edge is the number of vertices adjacent to both ends of the edge.
Theorem II.5.1. Let r and k be integers, such that r > k £ 2 . A graph H is the line graph of an (r,k)- semiregular bigraph, if and only if H has three properties* (1) Each vertex has degree r+k-2, (2) Each vertex is in an edge of degree r-2 and an edge of degree k-2 , and (3 ) The graph is not an induced subgraph of H.
A line graph of a BIBD is the line graph of the bigraph of varieties and blocks of a balanced incomplete block design.
Theorem II.5.2. Let v, k, b, r, be positive integers, such that r > k ^ 2 » bk = vr , and r(k-l) ~ ^(v-1 ) . A graph H is the line graph of a BIBD with these parameters, if and only if H has properties (l), (2 ), (3 ), and* (4) H has vr vertices, and
(5) Each edge of degree k-2 is in at most ^-1 four-cycles. For any positive integer m, the cocktail-party graph CP(m) is a complete graph on 2m vertices, minus m disjoint edges. By convention, CF(O) is the null-graph. Let G he a graph on p vertices. Let he non-negative integers. The generalized line graph H - L(G;a-,•*.,a ) is a graph composed of the line graph Jr L(G), and p cocktail-party graphs CP(a.,), ..., CP(a ). It Also, each vertex of L(G) that corresponds to an edge of G incident with the i-th vertex of G, is adjacent in H to each vertex of CP(a^).
Theorem III. . 3 . If Kv « is an induced subgraph of a generalized line graph H, then two of the three nonadjacent vertices are in the same cocktail-party graph.
Theorem I I I . If K^-Kg is an induced subgraph of a generalized line graph H, then the two nonadjacent vertices are in the same cocktail-party graph.
There are nine graphs, G^ - G^ , which cannot he induced subgraphs of a line graph. The graph G^ is and the graph G^ is K^-Kg.
Theorem III.5■2. None of the seven graphs Gg, G^ - G^ is an induced subgraph of a generalized line graph. Let G be a graph on p vertices. The ad.iacencv matrix of G is the p-square matrix A(G) with (i,j)-entry equal to 1 if vertex i and vertex j are adjacent.in G, otherwise 0. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. S Theorem IV.k .5. If a graph G has least eigenvalue ^ -2, then none of the three graphs Q, y or K^-LI of Pig. IV.^.1 is an induced subgraph of G.
Fig. IV.^.1. The graphs Q, K2 y and K^-LI
The base graph B(Htab) of a graph H is defined in Section 5 of Chapter IV.
Theorem IV.5.5. Let H be a graph, with an edge ab. If the least eigenvalue of H is ) -2, then the base graph B(Hiab) is an induced subgraph of CP(m)UK1 or the SchlSLfli graph.
Theorem IV.5.6. Let H be a graph, with an edge ab. Assume that each vertex of H is adjacent to a or b (or both), and that none of Q, K2,3* or an induced subgraph of H. If the base graph B(H*ab) is an induced subgraph of CP(m)LjK^ or the Schl&fli graph, then the least eigenvalue of H is ^ -2. A graph is exceptional if its least eigenvalue is greater than or equal to -2, but it is not a generalized line graph. An exceptional graph is represented as the graph of inner products of a set of vectors in the Lie algebra root system Eg. The root system is a subset Of Eg. Section 2 of Chapter V gives the steps in the construction of a graph H that is represented in a root system. An edge ab in H corresponds to two vectors with inner product (a,b) = 1 • Any other vertex of H is in a subset Xpj, YH , 2^, or T^, according as it is adjacent to b and not a, a and not b, both a and b, or neither a nor b respectively. A symbol X, Y, or Z, represents a vertex in
Xri or Zjj. A symbol W represents two adjacent vertice x^ in X jj and y^ in Y^.. A . sign (+ or -) represents a vertex in T^.
Theorem V.3.1. The only exceptional trees that are represented in E^ are the two trees in Fig.
Fig. V.3.4. The two exceptional trees in E^
Theorem There are fewer than 30*000 connected exceptional graphs represented in the root system E^. 10 A multigraph is a finite nonempty set of vertices, together with a family of edges. Loops are not permitted. The line multigraph N = L(M) of a multigraph M has vertex set corresponding to the family of edges of M. Two vertices of L(M) are joined by t = 0, 1, or 2 edges, if in M they correspond to two edges with t ends in common. A t-cluster in L(M) is the image of a t-bond in M. The cluster"reduction graph H of N = L(M) is the line graph of the underlying graph of M.
Theorem VI.4.3. A multigraph N is a line multigraph, if and only if* N has no t-bond for t > 2 , and none of the nine graphs forbidden in a line graph, nor the three multi graphs in Fig. VI.3 .2, is an induced sub-multigraph of N. (L (L £L Fig. VI.3-2. The three forbidden multigraphs
Theorem VI.5.2. A multigraph N has least eigenvalue greater than or equal to -2, if and only if* (1) N has no t-bond for t > 2 , (2) None of the three multigraphs of Fig. VI.3.2 is an induced sub-multigraph of N, and (3) The cluster-reduction graph H of N has least eigenvalue greater than or equal to -2. CHAPTER I
DEFINITIONS
1. Graphs A graph G = (V,E) is a finite nonempty set V of p elements called vertices, together with a set E of q edges, where an edge is an unordered pair of distinct vertices. By this definition, a graph has no multiple edges, no loops, and no directed edges. The edge on vertices v and w may be written e = vw » we say that the two vertices v and w are adjacent, and that the edge vw is incident with its ends, v and w. The degree degG (v) of a vertex v in a graph G is the number of vertices of G that are adjacent to v in G. A graph is regular if the degrees of all its vertices are equal.
Theorem 1.1. (First Theorem of Graph Theory) The sum of degrees in a graph is equal to twice the number of edges.
Proof. Each edge has two ends, so it contributes 2 to the sum of degrees.
11 1 2 The complete graph (or clique) is the unique regular graph with p vertices and degree p-1. We use the notation to denote the graph on four vertices that is complete except that one pair of vertices is nonadjacent. A bipartite graph (or bigraph) is a graph whose vertex set can be partitioned into two subsets, such that no two vertices in the same subset are adjacent. The complete bigraph n is the bigraph with m vertices in one subset, n vertices in the other subset, and each vertex in one subset adjacent to each vertex in the other subset. Pig. 1.1 shows a few of the graphs defined so far.
The gra phsFig. 1.1. The graphsFig. graphs K^, K^, K^-Kg, and ^
Two graphs are isomorphic. written G = H or G = H , if there is a bijection from one vertex set onto the other, that preserves adjacency and nonadjacency. A walk of a graph G is an alternating sequence of vertices and edges vQ, e^, v1( ..., v en , vn, that begins and ends with vertices, such that the edge e^ is incident with the vertices v. , and v., and v. 1 ^ v. , for all i - 1, 2, ..., n . The walk may also be denoted 13 v«0 v. 1 v„2 ... v_ n . The walk is ------closed if v~ 0 = v n . and is' ' open otherwise. A path is a walk such that all vertices are distinct. The graph that consists of a path with n vertices is denoted by Fn . An n-cycle is a closed walk such that the n vertices v^, ..., vn are distinct, and n ^ 3 . The graph that consists of an n-cycle is denoted by. Cn * The graph is called a triangle. l*f 2. The line grach of a graph Let G be a graph. The line graph of G is the graph L(G), whose vertices correspond to the edges of G; two vertices of L(G) are adjacent if they correspond to two edges of G that have one common vertex. If e = vw is an edge in G f then e corresponds to a vertex of L(G), with degree*
degL (G)(e ) s degG (v) + degQ (w) - 2 . A vertex v-^ of degree d^ in G corresponds to a clique in L(G), with d. vertices and ^-d. (d.-l) edges. JL jl x
Lemma 2.1. Let G be a graph, with p vertices of degrees dl* d2 * dp* 3111(1 q edees - The 1^-ne graph L(G) has q vertices and qL edges, where*
P P 1l = -1 + * r *1 • 1-1
Proof. The number of edges in L(G) is i^d.(d.-l) X X = £ H d-2 - ^ Z- d^ ~ ^ JL d^2 - q , using Theorem 1.1.
A vertex is isolated if it is not incident with any edge.
Lemma 2.2. Let G be a graph with no isolated vertices. Then G is connected, if and only if its line graph is connected. Proof* If G is connected, then any two vertices of G are joined by a path. Thus for any two edges, there is a path from an end of one to an end of the other. Any two vertices of the line graph L(G) correspond to two edges of G, and the path between the edges in G defines a unique path in the line graph. Thus L(G) is connected. Conversely, let the line graph L(G) be connected. Since G has no isolated vertex, each vertex of G is incident with an edge, for any two vertices of G, choose an edge incident with one, and an edge incident with the other. The two corre sponding vertices in L(G) are joined by a path. This path defines a unique path in G, which joins the two vertices of G.
Fig. 2.1.shows the ten connected graphs on 1 to 4 vertices, and their line graphs. Only two of the line graphs are isomorphic to each othert L(K^) = • The "line graph" of is the empty set of vertices, which we call the null-graoh.
K1 K2 K3 Kl,3 % V K2 k4 G L>YNI>n 0 K \
Fig. 2.1. Ten connected graphs and their line graphs 16
The next lemma shows that no graph other than is the line graph of more than one graph. A graph G is called the root graph of its line graph L(G).
Lemma 2.3. If G and G* are connected graphs with isomorphic line graphs, then either G and G' are isomorphic, or else they are and
Proof. If G has 1 to 4 vertices, then this follows from Fig. 2.1. Let G have more than four vertices. Let L(G) and L(G') be isomorphic. Suppose that a triangle of three vertices {a,b,c} in the line graph corresponds to ^ in G, and corresponds to in G*. There must be another edge d, as shown in Fig. 2.2. The vertex d in L(G) is adjacent to either one or three of the vertices *fa,b,c7« But d in L(G*) is adjacent to two of the vertices {a,b,c}. Hence L(G) is not isomorphic to L(G'), a contradiction. Thus L(G) determines G and G* uniquely. Therefore, G and G * are isomorphic.
Fig. 2.2. Odd and even triangles 17 A subgraph of a graph H, is a graph that consists of a subset of the vertices of H t and a subset of the edges of H that have both ends in that subset of vertices. An induced subgraph of a graph H, is a graph that consists of a subset of the vertices of H, and all of the edges of H that have both ends in that subset of vertices. A triangle of three adjacent vertices in H is called odd if some vertex of H is adjacent to an odd number of the three vertices.
The property of being a line graph is characterized in the following theorem. Statement (2) is due to Krausz (19^3)t and statement (3) to van Rooij and Wilf (1965). Statement (4) was discovered independently by G. Neil Robertson (unpublished) and Beineke (1968).
Theorem 2.*f. The following are equivalent. (1) H is a line graph. (2) The edges of H can be partitioned into complete subgraphs, such that no vertex of H is in more than two of the subgraphs.
(3) ^ 1 3 ^-s no^ 3X1 induced subgraph of H, and if two odd triangles have a common edge, then the subgraph induced by their vertices is K^. (**) None of the nine graphs G1 - G^ of Fig. 2.3 is an induced subgraph of H. 18
5
G G 7
Fig. 2.3* The nine graphs forbidden in a line graph
Statement (2) follows easily from the definition of a line graph. A proof of the equivalence of (1) and (3) is given by Harary (1969, pp. 74-77). This is the basis of a computer-oriented algorithm to detect a line graph and output its root graph (Lehot 1974). Statement (4) is obtained by drawing all graphs on 5 or 6 vertices that contain two odd triangles with a common edge. 19 3. The line graph of a tree A tree is a connected graph with no cycles. The line graph of a path Pn on n vertices, is the tree No other tree is a line graph, since a tree that is not a path must contain which is forbidden in a line graph by Theorem 2.*K
The following theorem characterizes a graph that is the line graph of a tree.
Theorem 3.1. Let H be a connected graph. Then H is the line graph of a tree, if and only if none of the following graphs is an induced subgraph of Hi (1) An n-cycle Cn for n £ 4 ,
(2) , or (3) % - K 2 .
Proof. Let G be a tree, with line graph H = L(G) . By Theorem 2.^, ^ is not an induced subgraph of H. Suppose that H contains Cn for n ^ 4 . Then G contains a corresponding cycle Cn , contrary to the assumption that G is a tree. Suppose H contains K^-Kg* Then the correspond ing edges of G form plus an edge, contrary to the assumption that G is a tree. Therefore, none of these graphs (1), (2), or (3) is sin induced subgraph of H. Conversely, let H be a connected graph with none of the graphs (1), (2), or (3) as induced subgraphs. Each one of the forbidden graphs G1 - G^ of Fig. 2.3 contains either ^ or K^-Kg. Thus H contains none of G^ - G^f so by Theorem 2.4-, H is the line graph of a graph G. By hypothesis (l), Cn for n ^ 4 is not an induced subgraph of H. Thus Cn for n ^ b is not an induced subgraph of G* Now assume that G has q edges, and contains C^. If q = 3 » then G is C^, and H is K^» which is also the line graph of the tree If q ^ ^ , then since H is connected, G has an edge incident with one vertex of the in G. This corresponds to a subgraph K^-Kg in H, contrary to (3). Therefore, G is a tree.
Recall that a triangle in a graph is called odd if some vertex of H is adjacent to an odd number of the three vertices of the triangle. A triangle is even if it is not odd. Adding one edge to a tree produces a graph that has exactly one cycle. The line graph of a graph with one cycle is characterized by the following theorem. 21 Theorem 3.2. Let H = L(G) "be a connected line graph, H not isomorphic to or to K^-Kg. Then G has exactly one cycle, if and only if eithert (1) H has exactly one induced Subgraph Cn for n ^ 4 , and H has no * even triangle, or (2) H has no induced subgraph Cn for n £ 4 , and H has
exactly one even triangle.
Proof. Let H = L(G) be a connected line graph. Let g be the number of n-cycles in G, for n ^ 3 . Let h be the number of induced n-cycles in H for n b , plus the number of even triangles in H. If G = K- « , then g = 0 , H = K-j , and h = 1 . If G is plus an edge, then g = 1 , H = K^-Kg » and h = 2 . These are the two excep tions in the hypothesis. For any other H, we will prove that g and h are either both 0, or both 1, or both ^ 2. To prove this, we will show that* if g ^ 1 then h 1 ; if g > 2 then h ) 2 i if h 1 then g 1 ; and if h £ 2 then g ^ 2 . The edges of an n-cycle in G for n ^ k , correspond to the vertices of an induced n-cycle in H. The edges of a triangle in G correspond to the vertices of a triangle in H. This triangle in H is even, since a fourth edge of G cannot have an end in common with one or three of the three edges of the triangle in G .
Thus if g ^ 1 then h ^ 1 * if g ^ 2 then h ^ 2 . 22 Conversely, the vertices of an induced n-cycle in H for n b t correspond to the edges of an n-cycle in G. It remains only to show that the vertices of an even triangle in H correspond to the edges of a triangle in G. let T be an even triangle in H, with vertices a, b, c. Since T is even, any other vertex of H that is adjacent to any of ■ a, b, c, must be adjacent to exactly two of a, b, c. Suppose that T in H does not correspond to a triangle in G. Then T corresponds to « in G. An edge of G that has a common end with two of a, b, c, must be one of the three edges d, e, or f of Fig. 3*1* If H had a vertex other than a, b, c, d, e, f, then (since H i s connected) some vertex of H would be adjacent to d, e, or f, and nonadjacent to a, b, and c. No edge of G has this property. Thus H has t = 3# A, 5» or 6 vertices, including a, b, c. If t = 3 , then H = , contrary to hypothesis. If t = A , then H = K^-Kg , contrary to hypothesis. If t = 5 or 6 , then h ^ 2 and g 2 . In summary, if h ^ 1 then g ^ 1 * if h £ 2 then g 1^. 2 . This completes the proof.
e
G
Fig. 3»1* An even triangle T in H = 1(G) 23 The eigenvalues of a graph Let A "be a p-square matrix of complex numbers. The characteristic polynomial of A is the determinant det(xI-A) (see Marcus and Mine 1965 ^or details). The eigenvalues of A are the p zerps of the characteristic polynomial. It follows that a complex number p is an eigenvalue of A, if and only if Av = pv for some nonzero vector v. Let Rn denote the n-dimensional vector space over the field R of real numbers. A real symmetric matrix has all eigenvalues non-negative, if and only if it is the matrix of inner products of a family of vectors in Rn (Marcus and
Mine 1965» P» 182). Let G be a graph on p vertices. The adjacency matrix of G is the p-square matrix A(G) with (i,j)-entry equal to 1 if vertex i and vertex j are adjacent in G, otherwise 0. Thus A(G) is a real symmetric matrix, with 0 on the main diagonal, and entries 0 and 1 elsewhere. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. The eigenvalues of any real symmetric matrix are real numbers. Thus the eigenvalues of a graph are real numbers. Eigenvalues would be most useful if the spectrum of eigenvalues uniquely identified the graph. Unfortunately, they do not, since there are nonisomorphic graphs with the same eigenvalues. For example, the two graphs in Fig. k.l 24- have the same spectrum of eigenvalues, (approximately) 2.709, 1.000, 0.194-, -1.000, -1.000, and -I.903. The two graphs are not isomorphic.
r
Fig. 4-.1. Two isospectral graphs
The eigenvalues of a p-square matrix are the zeros of a polynomial of degree p. For p ^ 5 # there is no solution in radicals to a general polynomial of degree n. Thus for a large number of vertices, there may be no algebraic expression for the eigenvalues of a graph. However, there are numeric methods for approximating the eigenvalues. For some graphs, properties of the graph have been used to find the eigenvalues exactly. 2 5 5. Literature review A class C of graphs is said to be characterized by a set of properties, if any graph with these properties belongs to the class C. Most characterizations are either geometric or spectral. A geometric characterization uses properties of the graph, such as degree, or the number of vertices adjacent to both ends of an edge. A snectral characterization uses the spectrum of eigenvalues of the adjacency matrix of the graph. Table 5»1 presents references to many graph character izations. They are grouped by qualities of the graphs. Spectral characterizations are marked with an asterisk (*). Many of these characterizations are discussed in the review articles by Cvetkovic (1971) and Bose (1973)- The strongly regular graph of the triangular associa tion scheme (Connor 1958) is isomorphic to the line graph of a complete graph, L(Kfl). Chang (1959) gave a geometric characterization of the line graph L(K^). Interestingly, there are just three graphs that have the correct geometric properties (they correspond to n = 8), but are not line graphs. The line graph of a complete bipartite graph, L(K „), m » it is edge-regular, i.e. each edge has the same degree. This graph was characterized by Shrikhande (1959 b), Moon (1963), and Hoffman (196*0. There is one exceptional graph. It has the correct properties with m - n = b , but is not L(K^ ^ ). 26 TABLE 5.1 CHARACTERIZATIONS OP GRAPHS STRONGLY REGULAR GRAPHS Triangular association scheme. TgCn), LfK^), unordered pairs. Connor (1958). Shrikhande (1959 a), Chang (1959. i960). Hoffman (i960 a, i960 b)*, Seidel (1967, 1968)*. Square lattice graph. L2(n), L(Kn n ). ordered pairs. Shrikhande (1959 b). Seidel (1967. 1968)*, Doob (1970 b)*. Net graoh. Lr (k). Bruck (1963). Bose (1973)* Partial geometry (r,k,t). Bose (1963), Benson (1966). Symplectic and orthogonal graphs. Shult (1972).
EDGE-REGULAR GRAPHS Tetrahedral graph. T^(n), unordered triples. Bose and Laskar (1967). Aigner (1969 a). Bose and Laskar (1970)*. Graph of unordered m-tuples. Tm (n). Dowling (1969)* Cvetkovi6 (1969)*. Cubic lattice graph, ordered triples. Laskar (1967). Dowling (1968), Aigner (1969 b). Laskar (1969 a)*, Cvetkovic (1970)*. Graph of ordered m-tuples. Enomoto (1973)* Line graph of complete bigraph. LtK^ ). Shrikhande (1959 b), Moon (1963). Hoffman (196*4-5. Doob (1969, 1970 a)*. Line graph of the bigraph of a protective plane. Dowling and Laskar (1967). Hoffman (1965)** Line graph of the bigraph of a symmetric BIBD. Aigner and Dowling (1972). Hoffman and Ray-Chaudhuri (1965 a)*. 27 TABLE 5*1— Continued REGULAR GRAPHS Line graph of the bigraph. of an affine plane. Rao and Rao (1969)» Laskar (1969 b, 1970), Pellerin and Laskar (1975)* Hoffman and Ray-Chaudhuri (1965 b)*. Line graph of the bigraph of a BIBD with /\ = 1 . Rao and Rao (1969). Doob (1975)** Line graph of* the bigraph of a BIBD* Chapter II of this dissertation. Doob (1971)*. Cameron et al. (1976)*. Line graph of an (r.k)-semiregular bigraph. Chapter II of this dissertation. Line graph of a regular graph. Sachs (1967)*. Regular graphs. Perron (1907)*, Frobenius (1912)*, Hoffman (1963)*» Finck (1965)**
OTHER
Line graphs. Krausz (19*0). van Rooij and Wilf (1965), Beineke (1968), Harary (1969), Lehot (196*0* Ray-Chaudhuri (1967)*, Hoffman (1968, 1970)*, Doob (1973)*, Cameron et al. (1976)*, Bussemaker et al. (1976)*. Bipartite graphs. Coulson and Rushbrooke (19**0)*, Finck and Grohmann (1965)** Graphs. Collatz and Sinogowitz (1957)*, Harary (1962)*, Sachs (1963)*, Cvetkovic (1971)*. Harary et al. (1971)*, Harary and Schwenk (197*0*, Schwenk (197*0*. Hoffman (1975)*. Planar graphs. Kuratowski (1930).
* Spectral characterization (eigenvalues). a,b * Two.articles by the author in the same year. 28 The line graph of the bigraph of a balanced incomplete block design (BIBD) was characterized by Aigner and Dowling
(1972), and by Rao and Rao (1969)* for certain parameter values. A new characterization, for other parameter values, is proved in Chapter II of this dissertation. Spectral characterizations of the line graph of the bigraph of a BIBD were given by Hoffman and Ray-Chaudhuri (1965 a,
1965 b), and Doob (1975). Line graphs were first defined by Whitney (1932). They were characterized by Krausz (19^3)» van Rooij and Wilf (1965), Robertson (unpublished), and Beineke (1968). Spectral characterizations are due to Ray-Chaudhuri (1967), and Cameron et al. (1976). The earliest theorem on the spectrum of a graph is attributed to two chemists, Coulson and Rushbrooke (19^0). They showed that for a bipartite graph, the spectrum (as a point set on the real line) is symmetric about the origin. Various spectral results were proved by Collatz and Sinogowitz (1957). This was followed by the work of Hoffman, Sachs, Seidel, and others. Kuratowski (1930) characterized the family of planar graphs. Two graphs are homeomorphic if'they can be obtained from the same graph by subdividing edges with vertices of degree two. A graph is planar if and only if it has no subgraph homeomorphic .to or y the two graphs of Fig. 5.1. Fig. 5 .1. The two Kuratowski graphs, and ^
For some graphs, properties of the graph have been used to find the eigenvalues exactly. Eigenvalues of graphs have been computed by Collatz and Sinogowitz (1957)*
Hoffman (1963* 19^, 1965t 1975) * Hoffman and Ray-Chaudhuri (1965 a, 1965 b), Harary and Schwenk (197*0» and Doob (1970 a, 1975). CHAPTER II
THE LINE GRAPH OF A BALANCED INCOMPLETE BLOCK DESIGN
1. Introduction In this chapter, a balanced incomplete block design (BIBD) is defined as a structure with certain parameters, v, k, b, r, The line graph of a BIBD is the line graph of the bipartite graph of the BIBD. We characterize, by five geometric properties, the line graph of a BIBD with r > k . Any graph with these five properties is the line graph of a BIBD. Previous authors have characterized the line graph of a BIBD under other parameter restrictions* r = - k (Shrikhande 1959 b ) , r = > k (Moon 1963, Hoffman 196*0, and r = k )> ), (Aigner and Dowling 1972). Of the BIBDs with k / r / , the line graphs have been characterized only for )\ ~ 1 and k < r < 2k (Rao and Rao 1969, Pellerin and Laskar 1975)* Among the values of (k,r,/0 that are not covered by their results are: (2,*+,l), (3,6,1), and (3,5,2).
30 31 Section 2 defines semiregular bipartite graphs and BXBDs. In Section 3, we characterize the line graph of* a BIBD such that r > k and X = 1 . Two lemmas on the parameters of a BIBD are proved in Section 4. The main result of this chapter is Theorem 5*2. We prove that a graph H is the line graph of a BIBD with r > k , if and only if H has five properties: (1) Each vertex has degree r+k-2, (2) Each vertex is in an edge of degree r-2 and an edge of degree k-2, (3) The graph is not an induced subgraph of H, (4) H has vr vertices, and (5) Each edge of degree k-2 is in at most ^-1 four- cycles * In Section 6, we discuss the parameter restrictions in geometric characterizations of the line graph of a BIBD.
Theorem 5*2 completes the series of characterizations begun by Shrikhande, as shown in Table 1.1.
TABLE 1.1 GEOMETRIC CHARACTERIZATIONS OP THE LINE GRAPH OF A BIBD
1 ^ >i < r ! > = r r < k Impossible (Fisher 1940) | Same as r > k j r = k Aigner & Dowling (1972) • Shrikhande (1959 b) j r > k Theorem 5*2 Moon (1963)1 Hoffman (1964>| 32
2. The line graph of a semiregular bigraph A graph is a finite nonempty set of vertices, together with a set of distinct edges, where an edge is an unordered pair of distinct vertices. Two vertices are adjacent if they are joined by an edge. The 1 ine graph L(G) of a graph G is the graph whose vertex set corresponds to the edge set of G, with two vertices of L(G) adjacent if they correspond to two edges of G with one common vertex. The degree of a vertex is the number of vertices adjacent to it. A graph is regular if all vertices have the same degree. The complete graph (or clique) is the graph with r vertices, each adjacent to each other. A bipartite graph (or bigraph) is a graph whose vertex set can be partitioned into two subsets,- such that no two vertices in the same subset are adjacent. A bigraph is (r .k)"semiregular if each vertex in one subset (called the v "varieties") has degree r, and each vertex in the other subset (called the b "blocks") has degree k.
Lemma 2.1. Let r and k be integers, such that r > k ^ 2 . A graph H is the line graph of an (r,k)-semiregular bigraph, if and only if, for any vertex x of H, the vertices adjacent to x form two cliques, of sizes r-1 and k-1, with no vertex in common and no edge joining them. Proof. If H is the line graph of a semiregular bigraph G, then a vertex x corresponds to an edge of G. It is adjacent to r-1 edges at one end, and to k-1 edges at the other end. Since two edges in G have at most one end in common, the two cliques in H have no vertex in common. Since G is bipartite, no vertex of the (r-1)-clique of x in H is adjacent to any vertex in the (k-1)-clique of x in H. Therefore, no edge of H joins the two cliques of x. This proves the lemma in one direction. Conversely, let H be a graph with the property for each x. Then x is in a unique r-clique.and a unique k-clique. Let A be the set of r-cliques of H, and let B be the set of k-cliques of H.
Let G be a graph whose vertex set corresponds to a U b , with two vertices of G adjacent if the corresponding cliques have one vertex of H in common. If two cliques have a vertex x in common, then they have exactly one vertex in common, because by hypothesis, the vertices adjacent to x form two cliques, of size r-1 and k-1, with no vertex in common. Since r / k , the sets A and B are disjoint. An edge of G joins a member of A with a member of B. Thus G is an (r,k)-semiregular bigraph. For x in H, no edge of H joins the (r-l)-clique of x to the (k-1)-clique of x. Therefore, H is the line graph of G. 3^ Let v, k, b, r, X be positive integers. A balanced incomplete block design (BIBD), with parameters v, k, b, r, X t is a set A of v varieties, together with a family B = of b blocks. such that each block is a set of k varieties, each variety:is in r blocks, and any two varieties are in X blocks together. The biarraph of a BIBD is the bipartite graph whose vertices are the varieties and the blocks of the BIBD. A variety and a block are joined by an edge if the variety is an element of the block. The line graph of a BIBD is the line graph of the bigraph of the BIBD. The line graph of a BIBD is sometimes called the flag graph, and its vertices called flags (a,q), where the variety a is in the block q. Two flags (a,q) and (a',q') are adjacent in the line graph, if and only if a “ a* or q = q* . The parameters of a BIBD must satisfy two equations.
Lemma 2.2. (Yates 1936) In a BIBD, bk = vr and r(k-l) = X (v—1) .
Proof. The number of flags (a,q) is bk = vr . For a fixed variety a, the number of pairs (c,q), such that varieties a and c are on block q, and a ^ c , can be counted in two different ways: r choices of q and (k-1) choices of c, or (v-1) choices of c and X choices of q. 35 The definition of a BIBD implies that r ^ £ 1 . In a BIBD with r = , each block contains each variety. If r / )[ , then we have Fisher's inequality.
Lemma 2.3. (Fisher 19^0) In a BIBD, if r , then b ^ v . 3 6
3. Nonedge degree at most 1 A nonedge is a pair of nonadjacent vertices. The degree of a pair of vertices (edge or nonedge) is the number of vertices adjacent to both. The four-cycle is the unique regular graph on four vertices of degree two. If an (r,k)-semiregular bigraph has no induced subgraph C^, then in its line graph, a nonedge has degree at most one. In particular, the bigraph of a BIBD with ^1=1 contains no four-cycle. In the course of this chapter, we will introduce thirteen properties, numbered (1) through (13)• Our characterization of an (r,k)-semiregular bigraph with no Cji^, uses properties (1), (2), and (6).
Lemma 3.1. Let r and k be integers, such that r > k ^ 2 . A graph H is the line graph of an (r,k)-semiregular bigraph with no C^f if and only if H has three properties* (1) Each vertex has degree r+k-2, (2) Each vertex is in an edge of degree r-2 and an edge of degree k-2, and (6) Each nonedge has degree at most 1.
Proof. Let H be a graph with properties (l), (2), and (6). Let xy be an edge of H. If two vertices y* and y" are both adjacent to both x and y, then the degree of the pair y', y" is at least 2. By (6), y* and y" are adjacent. Thus y and the vertices adjacent to both x and y must form a clique. The number of vertices in the clique is one more than the degree of the edge xy. By (2), one of these cliques has size r-1, and another has size k-1. By (1), these are all the vertices adjacent to x. By Lemma 2.1, H is the line graph of a semiregular bigraph.
A BIBD with = 1 is a semiregular bigraph with one block on each two varieties. We characterize its line graph by adding property (*0.
Theorem 3.2. Let v, k, b, r, be positive integers, such*' that r /' k ) 2 , bk = vr. , and r(k-l) = v-1 • A graph H is the line graph of a BIBD with these parameters = 1) if and only if H has properties (l), (2), (6), and: (ty) H has vr vertices.
Proof. By Lemma 3.1, H is the line graph of an (r,k)- semiregular bigraph G. Using (4), the set A of r-cliques of H partitions the vr vertices of H into v cliques of size r. Similarly, the set B of k-cliques of H partitions the bk = vr vertices of H into b cliques of size k. Thus A is a set of v "varieties," and B is a family of b "blocks We will show that each pair of varieties is adjacent to 38 exactly one block (X = 1). A variety a in A is an r-clique of vertices of H. The r vertices of the clique are edges, say Xj = (a,qj) , where q1 qr are blocks. A block q^ in B is a k-clique of vertices of H. These vertices are edges of G, say (a,qj) and (°23,q5). (ojj.qj).....
(ckj 'qj ) * The edges (a*) and ^ci2,q2^ are nonad0acent to each other, and both are adjacent to the edge (a^g)*
By (6), the pair ( c ^ q ^ is not an edge of G * Thus the r(k-l) varieties c* • are all distinct. Now A consists of 1J r(k-l)+l = v varieties, namely a and c^.. for i - 2,3» _. f..,k , and j = l,2,...,r.. The varieties a and c^.. have exactly one block incident with both, namely q.. This is J true for any choice of a in A. Therefore, G is the bigraph of a BIBD with = 1 , and H is the line graph of G.
Pig. 3.1.shows two graphs which have properties (1), (**•), and (6), but not (2), and are not line graphs. They are two members of an infinite sequence of graphs H*(r), r ^ 3 > which we define as follows. The graph H*(r) has r +r vertices: r cliques Ci of size r, and one set D of r nonadjacent vertices. In addition, the j-th vertex of D is joined to the j-th vertex of for all i and j from 1 to r. Since H*(r) contains &lfy it is not a line graph, one of these graphs, H*(3), was found by Pellerin and Laskar (1975). 3 9 H*(3) H*(*0
Fig. 3.1. Two graphs which are not line graphs 40
4. The parameters of a BIBD Theorem 3*2 characterizes the line graph of a BIBD with r > k and > = 1 . We compare this with previous characterizations. I n Theorems 4.1 and 4.2, the replication number r must be less than 2k.
Theorem 4.1. (Rao and Rao 19^9) Let bk = vr , r(k-l) = v-1 , and 3 ^ k < r < 2k-1 . A graph H is the line graph of a BIBD with these parameters ()i = 1), if and only if H has properties (1), (4), (6), andi (7) Each edge has degree either r-2 or k-2.
The same geometric properties, at a different parameter value, admit exceptional graphs.
Theorem 4.2. (Pellerin and Laskar 1975) Let bk = vr , r(k-l) = v-1 , and 2 ^ k < r = 2k-1 . Then k = 2 and r = 3 ■ Furthermore, a graph H has properties (1), (4), (c), and (7), if and only if either H is the line graph of a BIBD with these parameters - 1), or else H is one of seven exceptional graphs. k l
The following two lemmas fill in some gaps in the parameter ranges covered by the above theorems. The first lemma concerns an affine plane. which is a BIBD with r = k+1 and X ~ 1 •
Lemma . 3. In a BIBD, if r = k+1 , then X = X or ^ - r .
Proof. Since r = k+1 and bk = vr = vk+v , the block size k must divide v. The relation r(k-l) = ^i(v-l) can be rewritten as r - X = rk - )\v . Hence, k divides r-/». If r > X , then the quantity (r-)i)/k is a positive integer. Thus 1 ^ (r-^)/k , so that k ^ r-^i = k+l-/\ , where X is a positive integer. Therefore, ^ - 1 .
An integer r is congruent to c modulo k, r - c (mod k) , if r = ak+c for some integer a.
Lemma ^-.A. In a BIBD, if r = c (mod k) , then k divides c(c-/^) .
Proof. Let r = ak+c for some integer a. Since /lb = /Wr/k = ^lv(a+c/k) is an integer, the block size k must divide /Ivc. Now ^v = r(k-l) + X - -c + X (mod k) . Therefore, k divides c(c-^). U-2
In particular, if )i - 1 and r = 2 or -1 (mod k) ,
4 then ,k = 2 . We obtained Lemma ^.*4- by generalizing a similar lemma which Pellerin and Laskar (1975) proved for r = 2k-1 . Table ^.1 shows the parameter intervals covered by Theorems 3.2, ^.1, and *f.2. For the parameters 1 < < r and r > k+1 , we introduce a new property in Section 5*
TABLE ^+.1 THE LINE GRAPH OF A NONTRIVIAL, NONSYMMETRIC BIBD
)i-l and k=2 )i=l and k > 2 1 < > < r r = k+1 Pellerin & Rao & Rao Impossible Laskar (1975) (1969) (Lemma ^.3) r a k+2 Included in Impossible r > 2k-1 (Lemma ^.4) k+2 < r Impossible Rao & Rao < 2k-1 (1969) See Section 5 r = 2k-1 Same as Impossible r = k+1 (Pellerin & . Laskar, 1975) r > 2k-1 Theorem 3*2 1 ^3
5. No subgraph K^-Kg
The graph is the complete graph on four vertices, minus one edge. It is the unique graph with four vertices and five edges (see Fig. 5*1) •
Fig. 5*1* The graph K^-K2
Theorem 5.1* Let r and k be integers, such that r > k ^ 2 . A graph H is the line graph of an (r,k)-semiregular bigraph, if and only if H has three properties: (1) Each vertex has degree r+k-2, (2) Each vertex is in an edge of degree r-2 and an edge of degree k-2,,and (3) The graph is not an induced subgraph of H.
Proof. Let x be a vertex of H. If xy is an edge, then any two vertices adjacent to both x and y must be adjacent to each other, by (3). By (2), there is an edge xw of degree r-2, and an edge xy of degree k-2. By (1), the neighbors of x form two cliques, of order r-1 and k-1. By Lemma 2.1, the graph H is the line graph of an (r,k)-semiregular bigraph. 4/*
Recall that the graph of a BIBD is an (r,k)- semiregular bigraph such that each pair of varieties has exactly^ blocks in common. We characterize the line graph of a BIBD by adding conditions (4*) and (5)*
Theorem 5.2. Let v, k, b, r, X , be positive integers, such that r > k ^ 2 , bk - vr , and r(k-l) - )l(v-l) . A graph H is the line graph of a BIBD with these parameters, if and only if H has properties (1), (2), (3), and: (4) H has vr vertices, and (5) Each edge of degree k-2 is in at most ^-1 four-cycles.
Proof. The line graph of a BIBD has properties (1) - (5)* In fact, for property (5)i there are exactly ^-1 such four-cycles. Conversely, let H be a graph of vertices and edges, with properties (1) - (5)*- By Theorem 5 * 1 ► H is the line graph of an (r,k)-semiregular bigraph G. By (4), the set A of r-cliques of H partitions the vr vertices of H into v cliques of size r. Similarly, the set B of k-cliques of H partitions the bk = vr vertices of H into b cliques of size k. Thus there are v "varieties" in A, and b "blocks" in B. We will show that each pair of varieties in A is incident with exactly^ blocks in B. Fix a variety a, as shown in Fig. 5.2. If c is a variety Fig. 5*2. A four-cycle in the line graph of a BIBD incident v/ith at least one block q^ in common with a, then two vertices of H, denoted x = (a,q^) and y = (c,q^) , are adjacent. Since the block q^ is adjacent to k varieties, the degree of the edge xy in H is k-2. Let zi wi 130 811 edSe of sucl* that x, y, z^, w^^ is a four-cycle. Say that = (c,q^) , and that w^ - (a,q^) where q^ is a block distinct from q^. These four vertices of H, and the two varieties and two blocks that are their components, are shown in Fig. 5*2. By (5), there are at most )j-l edges z* w* in H, such that x, y, z., w. is JL JL J_ J_ a four-cycle. If a variety c is incident with at least one block in common with a, then c is incident with at most ?\ blocks in common with a. There are v-1 varieties c / a in A. Let nc be the number of blocks that are adjacent to both a and c. Thus the numbers nc are v-1 integers between 0 and }\. Now count in two different ways the number of pairs (c,q) such that both a and c are 46 varieties on the block q. The sum of n , over all c / a v in A, is equal to r(k-l) = X (v-1) . This sum has v-1 terms, 0 < n„ < X . Thus n_ must be equal to for each C v c ^ a in A. Therefore, G is the bigraph of a BIBD with parameters v, k, b, r, X ? "By Theorem 5*1» H is the line graph of G. 47 6. The line graph of a BXBD
Theorem 5*2 completes a series of characterizations, which "began with an article "by Shrikhande (1959 b) on the
square lattice graph, L(Kn i xi When Yates (1936) first defined BIBDs, he proved that bk = vr , and that r(k-l) = ^i(v-l) . By definition,
r£>i£l. If r=/\, then the bigraph of the BIBD is the complete bigranh K . , which has r varieties, k blocks, r , it and each variety adjacent to each block. The line graph L(]C v ) was characterized by Shrikhande r , it for r = k . There is one exceptional graph, which has the degree and edge degree for (v,k,^) = (4,4,4), but is not a line graph. Moon (1963) characterized the line graph of LfK^ jc), except for two cases, which were completed by Hoffman (1964). Now consider r > )\ . Fisher (1940) proved that r ^ k . If r = k , then the BIBD is called symmetric. The line graph of a symmetric BIBD was characterized by Aigner and Dowling (1972). There is one exceptional graph, with parameters (v,k,?\) = (7,4,2) . Finally, the case r > k is settled by our Theorem 5*2. These character izations are arranged according to their parameter restrictions in Table 1.1, at the beginning of this chapter. Their theorems may be stated as follows. 14- 8 Theorem 6.1. (Shrikhande 1959 b) Let v = k = r . A graph H has properties (1), (*0» and: (8) Each edge has degree r-2, and (9) Each nonedge has degree 2; if and only if either H is the line graph of Kr , r (the trivial symmetric BIBD), or H is the unique exceptional graph with r = ^ .
Theorem 6.2. (Moon 1963* Hoffman 196*0 Let r > v = k ^ 1 . A graph H has properties (1), (*0» (9)» and* (10) There are ^-kr(r-l) edges of degree r-2, and -§-rk(k-l) edges of degree k-2; if and only if H is the line graph of K (the trivial r i k . BIBD).
Theorem 6.3. (Aigner and Dowling 1972) Let X < k and ?\(v-l) = k(k-l) . A connected graph H has properties (1), (±0, (8), andi (11) Each nonedge has degree at most 2, (12) If (x,y) is a nonedge of degree 1, then there are k-^/ vertices adjacent to x and at distance 3 "to y, and (13) Each two vertices are at distance at most 3; if and only if either H is the line graph of a symmetric BIBD with these parameters, or else H is the unique exceptional graph with parameters (v,k,^) = (7,*f,2). CHAPTER III
GENERALIZED LINE GRAPHS
1. Introduction In this chapter, we prove some "basic facts about generalized line graphs. In Section 2, we show that the least eigenvalue of a line graph is greater than or equal to -2. Other graphs with this property, the generalized line graphs of Hoffman (1970), are defined in Section 3» In Theorem 2.4 of Chapter I, there are nine graphs which cannot be induced subgraphs of a line graph. In Section 4, we show that two of these can be induced subgraphs of a generalized line graph* ^ and K^-LI. In Section St we show that the other seven graphs forbidden in a line graph are also forbidden in a generalized line graph.
49 50 2. Cocktail-party graphs A graph is a finite nonempty set of p elements called vertices. together with a set of q edges, where an edge is an unordered pair of distinct vertices. The null-graph is an empty set of vertices* strictly speaking, it is not a graph. Let G be a graph with p vertices. The ad.iacency matrix of G is the p-square matrix A(G), such that the (i,j)-entry is equal to 1 if vertex i and vertex j are adjacent (joined by an edge), otherwise equal to 0 . The line graph L(G) of a graph G is a graph whose vertices correspond to the edges of G. Two vertices of L(G) are adjacent if they correspond to two edges of G with one common vertex. Let G be a graph with p vertices and q edges. The incidence matrix of G is the p-by-q matrix with (i,j)-entry equal to 1 if vertex i is incident with edge j, otherwise equal to 0. We multiply K on the right by its transpose matrix K to obtain the matrix equation* (2.1) K KT = A(G) + diag(d±) , where diag(di) is the p-square diagonal matrix with (i,i)- entry equal to the degree d^ of the i-th vertex of G. We multiply K on the left by its transpose to obtain* (2.2) KT K « A(L(G)) + 21 . where I is the q-square identity matrix. 51 Lemma 2.1. The least eigenvalue of a line graph L(G) is greater than or equal to -2 .
Proof. Let K be the p-by-q incidence matrix of a graph G. m The eigenvalues of the matrix product K K are non-negative for any real matrix K. By (2.2), the eigenvalues of the y adjacency matrix A(L(G)) are greater than or equal to -2 .
Michael Doob (1973) characterized the line graphs with least eigenvalue equal to -2. Recall that a cycle in a graph is a closed walk with all n ^ 3 of its vertices distinct. A tree is a connected graph with no cycles.
Lemma 2.2 . The least eigenvalue of a connected line graph H = L(G) is equal to -2, unless G is a tree, or G has one cycle, and that cycle is odd.
The complete graph Kp is the unique regular graph with p vertices and degree p-1 . For any positive integer m, the cocktail-party graph CP(m) is the unique regular graph with 2m vertices and degree 2m-2. Thus the cocktail-party graph is complete except for m nonadjacent pairs of vertices. By convention, CP(0) is the null-graph. For m > 2 , the least eigenvalue of CP(m) is -2. 52 3. Generalized line graphs Let G be a graph with vertex set l,...,p , and let a- a be p non-negative integers. The generalized line graph L(G*a^,.»•,ap ) is the graph that consists of the line graph L(G) and p cocktail-party graphs* CPCa^), ...» CP(ap), where each vertex of L(G) that corresponds to an edge of G incident with the i-th vertex of G, is joined in L(Gja^,...,ap) to each vertex of CP(a^). The generalized line graph was introduced by Alan J. Hofftnan.
Lemma 3.1. (Hoffman 1970) The least eigenvalue of a generalized line graph is greater than or equal to -2 .
As an example, Fig. 3*1 shows P^, the path on four vertices, together with a generalized line graph, L(Pj^jO,1,2,1) •
a<
1 f 0
2 1 CP(1) 23 3 t 2 CP(2 )
4 1 CP(1)
Fig. 3.1. A graph and a generalized line graph 53 Let H = L(Gja1i...,ap) be a generalized line graph. Let E be the vertex set of L(G) , corresponding to the edge set of G. Let C be the union of vertex sets of the cocktail-party graphs CP(a^). The E-neighborhood of a vertex c in C , is the set of vertices of E that are adjacent to c in H.
Lemma 3.2. The E-neighborhood of each vertex c in C is a clique.
Proof. The vertex c in CP(a^) is adjacent to the vertices of E that correspond to edges incident with the i-th vertex of G. This vertex is common to the edges of G. The corresponding vertices in the line graph L(G) are thus adjacent to each other. That is, they form a clique.
t Two vertices c and c' in C are C-connected if there is a path in C from c to c '.
Lemma 3.3. If two vertices in C are C-connected, then they are in the same CP(a^), and their E-neighborhoods are equal.
Proof. In the definition of generalized line graph, two vertices in C are C-connected only if they are in the same GP(ai). The E-neighborhood of each c in CPCa^ is the set 5^ of vertices of L(G) that correspond to the edges of the i-th vertex in G. Thus the E-neighborhoods of two vertices in the same CP(a^) are equal.
Lemma 3.^ . For each vertex c in C, there is at most one vertex in C that is C-connected to c and nonadjacent to c •
Proof. Let c be a vertex of CP(a^). If a^ = 1 , then c is not adjacent to any other vertex of C, so it is not C-connected to any vertex. On the other hand, if a^ > 1 , then c and the vertices of C that are C-connected to c, form CP(a^). Exactly one vertex of CP(a^) is nonadjacent to c . 55 4. Permitted, graphs Let H - L(Gia1 *•••»a-) be a generalized line graph. x p Let E be the vertex set of L(G), corresponding to the edge set of G. Let C be the union of vertex sets of the cocktail-party graphs CPCa^).
Lemma 4.1. The graph K„ „ can be an induced subgraph of a 1 * J generalized line graph.
Proof. The generalized line graph L ( P y 1,0,0) and L(F2 slf1) are shown in Fig. 4.1. The vertices of C are indicated by the letter "c." The edges of the induced subgraph It, « are indicated by thicker lines.
Fig. 4.1. -j in "two generalized line graphs
Lemma 4.2. The graph can be an induced subgraph of a generalized line graph.
Proof. The generalized line graphs L(K^ y 1,0,0,0), 0 ,2,0), L(P2;3 ,0), and L(P^;4) are shown in Fig. 4.2. The vertices of C are indicated by the letter "c." The edges of the induced subgraph K^-K2 are indicated by thicker lines. 56
c i
c
I
Fig. k*2* Kg-K-2 f°ur generalized line graphs
Theorem If ^ is an induced subgraph of a generalized line graph H, then two of the three nonadjacent vertices are in the same cocktail-party graph. Hence, they have the same neighborhood in H.
Proof. Let ^ be an induced subgraph of a generalized line graph H = L(G;a^i • • • ra ) . Let E be the vertex set of L(G), and let G be the union of vertex sets of CP(a^).
Label the vertices of as ^.3* Suppose that vertex w is in C . By Lemma 3.2, its E-neighborhood is a clique, so at least two of the other vertices are in C. Say that x and y are in C. Nov/ w, x, y, are in C and are C-connected, so by Lemma 3*3» their E-neighborhoods are 57 equal. Thus z is in C . But x, y f z are three nonadjacent vertices in the same CP(a^), which is impossible. Therefore, vertex w is in E. That is, w corresponds to an edge of G, which has two ends. Of the vertices x, y, z of H, there must be two that are associated with the same end i of w in G. Since these two are nonadjacent to each other, neither corresponds to an edge of G incident with the end i. Therefore, both correspond to vertices of CP(a^). By the definition of generalized line graph, tv/o nonadjacent vertices in CF(a^) have the same neighborhood in H.
Pig. ^.3 . The graph ^
Theorem If an in(3-uced subgraph of a generalized line graph H, then the two nonadjacent vertices are in the same cocktail-party graph. Hence, they have the same neighborhood in H.
Proof. Let be an induced subgraph of a generalized line graph H = L(G;a- ,...,a_) • Let E be the vertex set Jr of L(G), and let C be the union of vertex sets of CP(a^). 58 Label the vertices of K ^ K Z as in Fig. 4-.4. Case 1 . Vertex w is in E. Case 1.1. Vertex u is in C. By Lemma 3*2, the E-neighbor hood of u is a clique, so x is not in E. Now u and x are in C, and are C-connected, so by Lemma 3«3» their E-neighborhoods are equal. Vertex w in E is adjacent to u, but nonadjacent to x, a contradiction. Thus Case 1.1 is impossible. Case 1.2. Vertices u, v, y are in E. If x were in E, then L(G) would contain all of K^-Kg, which is forbidden by Theorem 2.^4* of Chapter I. Say that x is a vertex of CP(a^). Then u, v, y correspond to three edges of G with . common end-vertex i. The vertex w in E corresponds to an edge of G that has an end-vertex in common with u, v, y. Since w and x are nonadjacent in H, i is not an end-vertex of w. This is impossible in G. Therefore, Case 1 is impossible. Case 2 . Vertices w and x are in C. Case 2.1. Vertex u is in C. Then w and x are in C and are C-connected, so by Lemma 3*3 they are in the same CP(a^). Thus the theorem is true in Case 2.1. Case 2.2. Vertices u, v, y are in E. Say that x is in CP(a-), and w is in CP(a-). Then u, v, y correspond to X J three edges of G with common end-vertex i, and common end-vertex j. Thus i = j . Therefore, the theorem is true in Case 2 .2. This completes the proof. 59
Pig. Ur,U-, The graph
/ 60 6. Forbidden graphs Recall from Chapter I that the nine graphs of Fig. 5.1 are not induced subgraphs of a line graph.
Theorem 1.2.^. The following are equivalent.
(1) H is a line graph.
• « •
(4) None of the nine graphs G^ - G^ of Fig. 5*1 is an induced subgraph of H.
G G 5 / T \ /\ V \ ./S r8 G v / \
Fig. The nine graphs forbidden in a line graph
Lemmas ^.1 and k.2 show that the graphs Gi ~ 3 and G^ = can be induced subgraphs of a generalized line graph. We now apply Lemmas 3*2 - 3 .if, to show that the other seven graphs, G2 and G^ - G^ , are forbidden in a generalized line graph. 6l
Lei; H = L(G;a1 » • • • I ap) be a generalized line graph. Let denote a vertex of the vertex set E of L(G), and let "c" denote a vertex of G, the union of vertex sets of CP(a^).
Lemma 5.1. If K^-Kg is an induced subgraph of a generalized line graph H, then it has one of the five patterns of Pig. 5*2.
Proof. Each vertex of H is either in E or in C.. The possible assignments of the vertices of to E and G are shown in Figures 5*2 and 5-3* By Lemma 3-2 , the E-neighborhood of a vertex in C must be a clique. This eliminates the first two patterns of Pig- 5*3* By Lemma
3-3> if "two vertices in C are C-connected, then their E-neighborhoods must be equal. This eliminates the last two patterns of Fig. 5*3* Therefore, K^-IC, has one of the patterns of Pig. 5*2.
►/\ •
Fig. 5*2. K^-Kg in a generalized line graph
Big- 5*3- K4-K2 no"b in a Seneralized line graph 62 Suppose that Gg of Fig. 5*1 is an induced subgraph of a generalized line graph H. By Lemma 5*li "the K^-Kg in Gg must have one of the five patterns of Fig. 5*2. Let y be the other vertex of Gg. With pattern PI, if y is in E, then all of Gg is in L(G), contrary to Theorem 1.2.4. With pattern PI, if y is in C, then the E-neighborhood of y is not a clique, contrary to Lemma 3*2. Let w be the vertex c in pattern P2. With pattern P2, P3» or P4, if y is in E, then w is in C, but its E-neighborhood is not a clique, contrary to Lemma 3 *2 . With pattern P2 , P3» or P4, if y is in C, then w and y are in C and are C-connected, but their E-neighborhoods are not equal, contrary to Lemma 3*3* With pattern P5» if y is in E, then two vertices in C are C-connected, but their E-neighborhoods are not equal, contrary to Lemma 3*3. With pattern P5, if y is in C, then two vertices of C are C-connected to y, and are nonadjacent to y, contrary to Lemma 3.4. Therefore, Gg is not an induced subgraph of a generalized line graph.
■ c or c Fig. 5.4. Gg not in a generalized line graph 63 A -triangle of -three vertices in E is odd if there is a vertex of H (either in E or in G) that is adjacent to one or three of them. An odd triangle in L(G) must correspond to ^ in G. Thus a generalized line graph cannot contain in E, if both of the 'triangles in K^-K2 are odd. Suppose that G^ of Fig. 5*1 is an induced subgraph of a generalized line graph H. By Lemma 5«1, the in G^ must have one of the five patterns of Fig. 5*2. Let y and z be the other two vertices of G^. Any assignment of y and z to C or E with patterns P2 - P5 is contrary to Lemmas 3.2 - 3.4. Thus G^ has one of the patterns of Fig. 5*5i containing PI. The first pattern is contrary to Theorem 1.2.4. For the other two patterns, the four e ’s in Kij,“K2 f°rm ^wo triangles, so there is no choice of G, a^,...,a such that the generalized line graph H = L(Gia^ ap) contains the pattern. Therefore, G is not in H
c
Fig. 5*5* not in a generalized line graph 64
Suppose that of Fig. 5-1 is an induced subgraph of a generalized line graph H. By Lemma 5-1» *Wie K^-K2 in G^ must have one of the five patterns of Fig. 5*2. Any assignment of the other two vertices of G^ to C or E is contrary to Theorem 1.2.4 or Lemmas 3*2 - 3*^» except for the five patterns of Fig. 5*6. In the first three patterns, the K^-K2 in E forms two odd triangles, so no generalized line graph contains it. In the last two patterns, the triangle in E corresponds to three edges of G; two edges have a common end, and the third edge has an end not shared by either of the other two edges. This is impossible in G. Therefore, G is not in H.
Fig. 5*6. G^ not in a generalized line graph
Suppose that G^ of Fig. 5*1 is an Induced subgraph of a generalized line graph. By Lemma 5*1 $ "the K^-K2 in G^ has one of the five patterns of Fig. 5*2. Any assignment of the other two vertices of G^ to C or E is contrary to Theorem 1.2.4 or Lemmas 3*2 - 3.4, except for the three patterns of Fig. 5*7* In the first two patterns, the 6 5 Kj^-Kg in E has two odd triangles, so no generalized line graph contains it. In the third pattern, the four •'s correspond to four edges of G with one common end. The two c*s are joined to only two of the •'s, which is impossible. Therefore, G^ is not in H.
Pig. 5*7 ■ .Gg not in a generalized line graph
Suppose that Gy of Fig. 5*1 is an induced subgraph of a generalized line graph H. By Lemma 5*1* the in Gy must have one of the five patterns of Fig. 5*2. Any assignment of the other two vertices of Gy to C or E is contrary to Theorem 1.2.^ or Lemmas 3*2 - J.k. Therefore, Gy is not in H. Suppose that Gg of Fig. 5*1 is an induced subgraph of a generalized line grah H. By Lemma 5*1« the in Gg must have one of the five patterns of Fig. 5.2. Any assignment of the other two vertices of Gg to C or E is contrary to Theorem I.2.A or Lemmas 3.2 - 3 *A, except for the two patterns of Fig. 5-8 • The Kj^-Kg in E has two odd triangles. Therefore, Gg is not in H. 66
Pig. 5*8• Gg not in a generalized line graph
Finally, suppose that G^ of Fig. 5*1 is an induced subgraph of a generalized line graph H. By Lemma 5*1> the Kj^-K^ in must have one of the five patterns of Fig. 5*2. Any assignment of the other two vertices of G^ to C or E is contrary to Theorem 1.2.4- or Lemmas 3.2 - 3*^* Therefore, G^ is not in H. We have proved the following.
Theorem 5.2. None of the seven graphs G2, G^ - G^ of Fig. 5*1 is an induced subgraph of a generalized line graph. CHAPTER IV
GRAPHS WITH LEAST EIGENVALUE > -2
1. Introduction This chapter presents the theory of graphs with least eigenvalue greater than or equal to -2. In Section 2, we show that a graph with least eigenvalue greater than or equal to -2, is the graph of inner products of a set S of vectors in Rn . Each vector in S has length >/2* Tw° distinct vectors in S have inner product 0 or 1. The vertices of the graph of inner products correspond to the vectors in S. Two vertices are joined by an edge, if the corresponding two vectors have inner product 1 . More generally, a set S* of vectors of length \/z in Rn , with inner products +1, 0, or -1, has an "edge-signed graph of inner products." The vertices of this graph correspond to the vectors of S*. An inner product of -1 is indicated by a negative edge (a dashed line). An inner product of +1 is indicated by a positive edge (a solid line).
6? 68 Cameron, Goethals, Seidel, and Shult (1976) proved that an indecomposable set S* of vectors of length \/2 in Rn , with inner products +1, 0, or -1, is a subset of a Lie algebra root system Dn or Eg. Their theorems are stated in terms of the line generated by a vector, but the proofs use the vector itself. In Section 3* we restate their results in terms of vectors. The structure of the set S* is given by the base graph B(S*:ab), with respect to a pair of vectors (a,b) = 1 . Section k deals with the root systems Dn and Eg. In Section 5» we define the base graph B(Hiab) of a graph H, with respect to an edge ab. Even if H has an eigenvalue less than -2, its base graph is well-defined, provided that none of the three graphs Q, Kg y or K^-LI is an induced subgraph of H. Assume that each vertex of H is adjacent to a or b (or both). We prove that the least eigenvalue of H is greater than or equal to -2 , if and only if the base graph B(H:ab) is an induced subgraph of either the graph CP(m)(jK1> or the Schiafli graph. 69 2. The edge-signed graph of inner products Let R denote the field of real numbers. A vector c in Rn is an ordered n-tuple (c^f•••fc ), such that each c^ is in R. The inner -product of two vectors c and d in
Rn is: n (c,d) = Z c. d. . 1 = 1 1 1 ______The length of a vector c is || c || = \/(c(c) . The Gram matrix of a set of vectors is the matrix of inner products. That is, the (i,;j)-entry is equal to the inner product of the i-th vector and the j-th vector in the set.
Lemma 2.1. A real symmetric matrix has all eigenvalues non-negative, if and only if it is the Gram matrix of a set. of vectors in Rn .
Proof. (Marcus and Mine 1965* PP* 182-18*0.
Let H be a graph with least eigenvalue greater than, or equal to -2. Then the matrix C = A(H) + 2 1 is a real symmetric matrix with all eigenvalues non-negative, so it is the Gram matrix of a set of vectors. The length of each vector is \/z% since each entry on the main diagonal of C is 2. Any two vectors have inner product 0 or 1. 70
Let S be a set of vectors of length s/2 in Rn , with inner products +1, 0, or -1. Two vectors are at an angle of 90° (or orthogonal) if their inner product is 0. Two vectors are at an angle of 60° if their inner product is +1. Two vectors are at an angle of 120° if their inner product is -1. We define an edge-signed graph to be a graph with two kinds of edges, positive and negative. The edge-signed graph of a set S of vectors has vertex set S. Two vectors are joined by a positive edge (solid line) if their inner product is +1, by a negative edge (dashed line) if their inner product is -1, and by no edge if their inner product is 0 . Two vectors are thus ad.iacent if their inner product is +1 or -1 (andle of 60° or 120°), and orthogonal or non-ad.iacent if their inner product is 0 (angle of 90°). If three vectors are mutually adjacent, then either 0, 1, 2, or 3 of ‘the three inner products are +1. These four possibilities are shown in Fig. 2.1. If 0 or 2 of the inner products are +1, then the three vectors lie in a plane. If 1 or 3 of the inner products is +1, then the three vectors do not lie in a plane. We call three adjacent vectors a star if they lie in a plane, or a cone if they do not. Stars in R^ Cones in R^
Number of positive edges Edge-signed graph
Vector diagram
Fig. 2.1. Three adjacent vectors in a star or cone
A set of vectors is star-closed if, with any two vectors a and b with inner product +1 or -1* it contains a third vector c such that {a,b,c} is a star. That is, if (a,b) = +1 , then c ~ + (a-b) . If (a,b) = -1 , then c = + (a+b) . A set of vectors in Rn is decomposable if it is the union of two nonempty subsets, in orthogonal subspaces. A set is indecomposable if it is not decomposable. Let S* be an indecomposable star-closed set of vectors of length \/2 in Rn , with inner products +1, 0, or -1. Let a and b be vectors in S* such that (a,b) 0 . Without loss of generality, (a,b) “ +1 . That is, if (a,b) = -1 , then we replace b in S* by -b. 72 Since S* is star-closed, either a-b or -(a-b) is in S*. Without loss of generality, let (2 .1) c = a-b be in S*. Let X be the set of vectors in S* that are orthogonal to a, but not to b. Without loss of generality, (b.x^) = -(c,x1) = +1 , for each x^ in X. Fig. 2.2 shows the edge-signed graph of a, b» c, and x^.
c
Fig. 2.2. The edge-signed graph of a,b,c,x^
Let Y be the set of vectors in S* that are orthogonal to b, but not to c. Without loss of generality, (a,y2) = (c,y2) = +1 . for each y2 in Y. Fig. 2.3 shows the edge-signed graph of a, b, c, and y2«.
Fig. 2.3* The edge-signed graph of a,b,c,y2 73 Let Z be the set of vectors in S* that are orthogonal to c, but not to a. Without loss of generality, (a.z^) = (b,z^) = +1 , for any in Z. Fig. 2.*J- shows the
edge-signed graph of a, b, c, and z^.
c
Fig. 2 .^. The edge-signed graph of a,b,c,z^
If S* contains a vector x^ in X, then S* contains (2 .2 ) y± = c+x^^ in Y, since S* is star-closed. Similarly, if S* contains y^ in Y, then S* contains (2.3) = a-yi in Z. Also, if S* contains z^ in Z, then S* contains (2.*0 x^ = in X. VJe calculate the inner product (x^,y^) = (x^,c+x^) = (x^,c) + (xi ,xi) = -1+2 = +1. Similarly, (yi,zi) = -1 , and (x^.z^) = -1 . Thus is a cone, which we call cone i of S*. Fig. 2.5 shows the edge-signed graph of inner products of a, b, c, x-, y ., and z.. J* * J# 74
c
Fig. 2.5. The edge-signed graph of a,b,c,x-,y.,z. •» X X
Finally, let T be the set of vectors id S* that are orthogonal to a, b, and c.
Lemma 2.2. If x^ and x2 are two vectors in X, then (x^.Xg) is either 0 or 1 .
Proof. The inner product (b,x1 ) = (b,x2) is equal to 1, and (x1 ,x2 ) is either -1, 0, or 1. Suppose that (x^.Xg) = -1. Then x2 = b-x^ - z^ . But a is adjacent to z^ and not adjacent to x2 , a contradiction. Therefore, (x^Xg) is not equal to -1.
If (xlfX(.) = 1 , then by equations (2.l)-(2.4), the vectors of cone 1 and cone 5 must be joined "alike," namely* x^ to x^, y^ to y^, and z^ to z^. Since S* is star-closed, the difference vector is in S*. 75 Lemma 2.3* If (x^,x^) = 1 , then the difference vector t ^ - x-^-x^ is orthogonal to a, b, and c. It has inner product +1 with x ^ , y^, and z,-. It has inner product -1 with Xtj, y^, and z^ .
Proof. All nine calculations are easily made, using the linearity of inner product. For example, (a,t^-) = 15 (a,x1 ) - (a,x^) = 0 - 0 - 0 .
We call this the Type 1 connection of cones 1 and 5» as shown in Fig. 2.6.
15 \
x
Fig. 2.6. Type 1 connection, with (x^,x^) - 1
On the other hand, if (x^,x2) - 0 , then cone 1 and cone 2 have the Type 0 connections, shown in Fig. 2.7. The vectors of cone 1 are joined to the "unlike" vectors of cone 2. For example, z^ is adjacent to x2 and to y2, but z^ is nonadjacent to z2 » Pig. 2.7* Type 0 connection, with (x^Xg) - 0
Lemma 2.*K If (x^,x2) = 0 f then there is a vector x^ in X , orthogonal to x^ and x2 . Any vector Xj^ in X \ fxlfx2,x^] is adjacent to exactly two of the three vectors x^, x2 , x^.
Proof. By (2.2) and (2.4), we have* (y^»z2) - (x.^+c.b-Xg) = 1+0-1-(-1) = 1 . Since S* is star-closed, it contains the vector x^ = “y i+z2 ' This x^ is orthogonal to a, x ^ and Xg. It is adjacent by a positive edge to b, and adjacent by a negative edge to c. Hence', x^ is in X. For any other x^, the inner product of x^ with y1 is 0 or -Is with x-j it is 1 or 0. Thus 0 = (x^-x^+x^) - (x^y^) - (x^,Zg) + (x^,x^) can only be either 0+0+0 , or 0-1+1 , or -1+0+1 . It follows that x^ is adjacent to x 1 and x2 , or to x 1 and x^» or to x2 and x^, respectively. 77 3. The base graph of a set of* vectors Cameron, Goethals, Seidel, and Shult (1976) characterized the graphs with least eigenvalue ^ -2 as graphs "represented by subsets of the root system" Dn or
Eg (Cameron et al. 1976, pp. 3 0 6 , 315-320). The results in their paper are stated in terms of a set of lines in Rn , which pass through the origin. In their proofs, a vector of length \/2. is chosen along each line in the set. For each line, there are just two such vectors, in the two opposite directions along the line. A subset of lines represents its graph of inner products, provided that the vectors are chosen so that all inner products between pairs of vectors are non-negative. In this section, we state their results in terms of vectors, not lines. In fact, the lines are not necessary in the proofs. As in Section 2 , let S* be a star-closed indecomposable set of vectors of length >/2 in Rn , with inner products +1, 0, or -1. Let a and b be vectors in S*, such that (a,b) = +1 . Define c, X, Y, Z, arid T, as in Section 2. Define the base graph B = B(S*iab) of S*, with respect to ab, to be the graph of inner products of the subset X. By Lemma 2.2, (x*,x.) / -1 for any x. and x. in X. The direction of vectors in S* was chosen in Section 2, such that the base graph has no negative edges. 78
The importance of S* being star-closed is shown in Table 3.1. Let S be a set of vectors of length\/2 in Rn , with inner products +1, 0, or -1. Let a and b be in S, with (a,b) = +1 . Three typical vectors x^, y2, z^ in S have inner products with a and b as shown in Figures 2.2, 2.3, and 2.^. If S is a subset of a star-closed set S*, then S* contains cone 1, {x1(y^*z^}» with inner products as shown in Fig. 2*5* Similarly, S* contains cone 2 and cone 3• The vectors x^» x2» x^, are vertices of the base graph B(S*:ab). The inner product (x*,x.) = 1 or 0 determines all adjacencies between cone i and cone j, as in Fig. 2.6 (Type 1) or Fig. 2.7 (Type 0). In particular, the adjacencies of x^, Xg, x^ in B(S*:ab) determine the inner products of x^, y2 » z^ in S.
TABLE 3.1 THE BASE GRAPH OF A SET S OF VECTORS
Vectors in S Vectors in S* a,b Vertices of B(S*iab)
b,c 79 Now Lemma 2 . Ur is stated in terms of the base graph, as follows.
Lemma 3.1. The base graph B = B(S**ab) satisfies two conditions * (3.1) If x^ and Xg are nonadjacent vertices of B, then there is a unique vertex x^ of B, nonadjacent to
both x^ and x2 i and (3.2) Any x^ in V(B)\{XjiXg.x^} is adjacent to exactly two of x^,x2 ,x^
VJhat graphs B satisfy conditions (3«1) and (3»2)? For any n £ 1 , the complete graph Kn does trivially, since there are no nonadjacent vertices. Also, the graph CP(m)U consisting of a cocktail-party graph and one isolated vertex, satisfies the conditions. Each vertex is nonadjacent just to its mate and to the isolated vertex. A third example is the line graph L(K^ ^). Fig. 3.1 shows L(K„ „) drawn in full, and also in a schematic j * j notation. All of the vertices along a double bar are adjacent to each other. If two vertices are nonadjacent, then the third vertex nonadjacent to both is the unique vertex of L(K^^) that is not in the same row or column as the first two vertices. • 80
i i' ►
If
Pig. 3.1. The line graph L(KQ „) j % j
A fourth example is the line graph L(K^)t shown in Pig. 3.2. This graph consists of six cliques. Any vertex is the intersection of two cliques. A nonadjacent vertex is the intersection of two other cliques. Their unique nonadjacent third vertex is the intersection of the other two cliques of L(K^). 81
Fig. 3.2. The line graph Ii(K^)
The fifth, and final, example of a graph satisfying properties (3.1) and (3.2) is the Schldfli graph. Its twenty-seven vertices correspond to the lines on the general cubic surface (Schlflfli 1858). The graph is also described as the polytope 221 by Coxeter (19^0, 1962 p. 211, and 1972* pp. 118-125). Segre (19*1-2) represented the Schl&fli graph as the set of ordered triples (xyz) on four symbols {o,l,2,3] , such that each triple has exactly one 0. Two triples are adjacent if they agree in either none or two of the three positions. In this representation (see Fig. 3*3) i the subgraph in the plane z - 0 is isomorphic to L(K« «), and j » j the subgraph in the region z ^ 1 is isomorphic to L(K^). 82
130 230 330
031
120 220 320
033, 021
022 110 210 310
023 Oil
012
013 2Q1 301101
102 202 302
103 203 303
Fig. 3.3. Segre coordinates for the Schl&fli graph. Two vertices are adjacent if they agree in 0 or 2 coordinates. All edges are drawn for vertex 110j other vertices are similar. 83 Cameron et al. (1976, p. 309) proved that no other graphs satisfy properties (3*1) and (3*2).
Lemma 3.2. Let B he a graph that satisfies conditions (3*1) and (3.2). Then B is either K , CP(m)U K1, L(K_ in x J t J L(K^), or the Schlfifli graph.
This is the content of Lemma 2.9 and 2.10 of Cameron et al. In their proofs, the inner products of a, b, c, x - , y *, and z. are as shown in Fig. 3 .^. This X J* is the same as our edge-signed graph in Fig. 2.5i except that the direction of a is opposite, and the direction of is opposite. Their convention is suitable for an existence proof, but our convention is useful for both existence and construction. Also, Cameron et al. say that two vectors are adjacent if the inner product is 0 . Thus, their "graph on Fa" is the complement of our base graph on X*
C
x.
Fig. 3»^» Inner products of Cameron et al. (1976) 8^ Now Lemmas 3.1 and 3*2 produce "the following result on the base graph of a set of vectors in Rn .
Theorem 3.3* Let Sw be an indecomposable star-closed set of vectors of length \fz in Rn f with inner products +1, 0 F or -1. Let a and b be two vectors in S*, with inner product (a,b) = +1 . Then the base graph B(S*:ab) is either K^. CPCnOijKi* L(K6) , or the Schiafli graph.
Proof. This is Theorem 2.13 o f Cameron et al. (1976). 85 Root systems In Section 3t we defined the base graph of an indecomposable star-closed set S* of vectors of length in Rn , with inner products of +1, 0, or -1. Theorem 3.3 implies that the base graph is an induced subgraph of either CP(m)\jK^ or the Schldfli graph. This leads to
a characterization of such sets of vectors, in terms of the root systems of Lie algebra. A root system is a finite set P of nonzero vectors spanning Rn , such that: (1) If r ando(r are in F, where oi. is in R, then o<, = + 1 ; (2) If r and s are in F-, then 2(r,s)/(r,r) is an integer; and (3) If r and s are in F, then the "reflection" wr (s) = s - 2(r,s)r/(r,r) is in F. Root systems are studied in Chapter 2 of Carter (1972). Cameron et al. (1976) use three families of root systems in the characterization of graphs with least eigenvalue £ -2. To define them, let Bn = {e^ en] be an orthogonal basis for Rn . The root system An , n ^ 1 , is the set of vectors ± (e- - e.) , for 1 $ i < j ^ n+1 , e. 6 Bn+1 . J- J X The set A spans an n-dimensional subspace of B , since n n+1 all of the vectors of An are orthogonal to the vector 27 e•• i=*l 1 86 The root system Dn , n ^ k , is the set of* vectors + (ei + e^) , for l ^ i < j < ^ n , e i £ 3n .
Thus An is a subset of Dn+i» The root system Eg consists of the vectors of Dg, and the vectors 8
where a(1. = + 1 , T * T 1 . = 1 ,
The root system E^ is the set of vectors in Eg that are orthogonal to any one of its vectors. The root system Eg is the set of vectors in Eg orthogonal to any two non- orthogonal vectors. Notice that the negative of any vector in a root system is also in the root system. Cameron et al. (1976) define the root system to be the set of lines generated by the vectors listed above. The number of lines in An is ^n(n+l). The number of lines in Dn is n(n-l). The number of lines in Eg, E^, Eg is 120, 63, 36 , respectively. We follow Carter (1972) in defining the root systems to be the set of vectors, not the set of lines generated by the vectors. In Chapter III, Lemma 3-1 states that the least eigenvalue of a generalized line graph is ^ -2. With the notation of root systems, we now present a proof of that Lemma. 87 Lemma III*3.1. (Hoffman 1970) The least eigenvalue of a generalized line graph is greater than or equal to -2.
Proof* Let G be a graph on p vertices. Let a-, a be non-negative integers, and let n = 2£_ (1+a*)/ x .
1 = 1 - The generalized line graph LCGja^,..•,ap) is the graph of inner products of the following set S of vectors in Dn * Let {eid. 1 , l ^ i ^ p j be an orthononnal basis for Rn . Let S consist of the vectors: (e- « + e. J , for each edge (i ,j) of G, and i , U J ) U
(ei,o + ei.*> 3 1 1 3 / Theorem 4.1. Let S* be an indecomposable star-closed set of vectors of length ]/2. in Rn , with inner products +1, 0, or -1. Then the set of + the vectors of S* is one of the root systems An , Dn , E6 , E? , or Eg, Proof. This is Theorem 3,5 of Cameron et al. (1976). The base graphs of these root systems are* B(An *ab) = g , B(Dn *ab) = CP(n-3)UK1 , B(E6 *ab) * L (K3 ,3> . B(E? :ab) = l (K6) , and B(Egiab) is the SchlSfli graph. 88 Theorem 4.2. A connected graph H has least eigenvalue greater than or equal to -2, if and only if H is the graph of inner products of a set S of vectors in a root system Dn or Eg. Proof. Let H he a connected graph with least eigenvalue £ -2. Then C = A(H) +21 is a real symmetric matrix with least eigenvalue non-negative. By Lemma 2.1, C is the Gram matrix of a set S of vectors in Rn . Each vector in S has length \/Zt and two vectors in S have inner product +1 or 0. Since H is connected, S is indecomposable. Let S* be an indecomposable star-closed set containing S. By Theorem 4.1, + S* is one of the root systems An , Dn , Eg, E y t or Eg. But An_i C Dn » ^d E7 C— E 8 ' Conversely, let S be a subset of Dn or Eg, with all inner products non-negative. Then the matrix C of inner products of S has least eigenvalue non-negative. The graph H of inner products of S has adjacency matrix A(H) = C - 21 . Therefore, the least eigenvalue of H is greater than or equal to -2. Let A be a p-square matrix, with rows and columns indexed by the set {l,2,...,p} . The principal submatrix of A induced by a subset of {l,2,...,p} , is the matrix of entries a ^ such that i and j are in the subset. 89 An induced subgraph of a graph G, is a graph that consists of a subset of the vertices of G» and all of the edges of G that have both ends in the subset. Thus, the adjacency matrix of an induced subgraph of a graph is a principal submatrix of the adjacency matrix of the graph. The eigenvalues of a matrix and a principal submatrix are related by Cauchy's inequalities. Theorem 4.3. Let A be a p-square hermitian matrix, with eigenvalues ^ ^ Ap • Let Z be a k-square principal submatrix of A, with eigenvalues ^ ... ^ • Then ^s ^ hs ^ ^p-k+s * for s = 1,...,k . Proof. (Marcus and Mine 19^5» P* 203.) Corollary 4.4. The least eigenvalue of a graph is less than or equal to the least eigenvalue of any induced subgraph. Theorem 4.5. If a graph G has least eigenvalue £ -2, then none of the three graphs Q, or K^-LI of Pig. 4.1 is an induced subgraph of G . 90 Proof. The eigenvalues of the 1^3 connected graphs on 1 to 6 vertices are tabulated in Appendix A. The least eigenvalue of Q is approximately -2.136* The least eigenvalue of Kg ^ approximately -2.^9* The least eigenvalue of K^-LI is approximately -2.177. By Corollary ^.4, none of these three graphs is an induced subgraph. Pig. if-.l. The graphs Q, Kg and K^-LI 5. The base graph of a graph Let H be a connected graph, with adjacent vertices a and b. Assume that none of Q, Kg ^ , or K^-LI is an induced subgraph of H. Let XH be the set of vertices of H (other than a) that are adjacent to b, and nonadjacent to a. Let Y^. be the set of vertices of H (other than b) that are adjacent to a, and nonadjacent to b. Let Z^ be the set of vertices of H that are adjacent to both a and b. Let Tjj be the set of vertices of H that are adjacent to neither a nor b. Thus the vertex set of H is {a,b)Uxjj^YH ^ ZH ^ TH * Lemma 5.1. A vertex y in is adjacent to at most one vertex x in X^. Proof. Suppose that y in Yjj is adjacent to both x and x* in XH. Then or 501 induced subgraph of H, as shown in Pig. 5*1• This contradicts the assumption that Kg 3 and K^-LI are not induced subgraphs of H. Fig. 5»1* H does not contain K0 „ or K--LI 9 J D Let Y^* te the set of vertices in Y^ that are not adjacent to any x in X. Let V he a set of vertices, with a bijection v.i xj{Uxn*U ± V . The base graph B(H*ab) of H, with respect to the edge ab, has vertex set V = (vCx-^) i x ± € XH}U { v(y^) t y^ € YH '\ U (v (zk ) 1 2 The adjacencies in B(Hiab) are defined in the following rules (1) - (6). (1) If Xii and x- j are vertices in X„, n then make v(x*) i and v(x.) adjacent in B(Hiab) if and only if x- and x. are J J adjacent in H. (2) If y^ and y.. are vertices in Y^*, then make v(y^) and v(y-) adjacent in B(Hiab) if and only if y^ and y. are J J adjacent in H. (3) and Zj are vertices in zH. then make v(z^) and v(z.) adjacent in B(H:ab) if and only if z* and z- are J U adjacent in H, (^l')lf is in Xjj and y.» is in v , then x^ and y^ must be nonadjacent in H, by the definition of Y^1. We n make v(x^) and v,(y.) adjacent in B(Hiab) . J (5 ) If i*s in elxicL z* j is m ZH , then make v(x^) and v(z.) adjacent in B(H*ab) if and only if x- and z. are J J nonadjacent in H. (6 ) If y ± is in YH * and z^ is in Z^, then make v(y^) and v(z^) adjacent in B(Hiab) if and only if y^ and z. are J nonad.iacent in H. 93 If H is "the graph of inner products of a set S* of vectors, then rules (1) - (6) are designed to make a vertex v(xi), v(yi), or v(zi) adjacent in B(H:ab) to a vertex v(x-), v(y-), or v(z.), if and only if the vectors J J J x- and x- in S* have inner product +1. i j The correspondences between vertices x^# y^i z^ of H, vertices vCx^), v(y2), v(z^).of.B(H*ab), and vertices x ^ , Xg, Xj of B(S**ab), are shown in Table 5*1 • TABLE 5.1 THE BASE GRAPH OP A GRAPH H Vertices of Vertices Vectors Vectors Vertices of B(Hiab) of H in S in S* B(S*iab) — a a a — — b b b, c —— vCxj^) X 1 * XH x± i X X l ,yl ,zl € X x 2 € X v(y2 ) y2 * YH y2 s * X2 ,y2 ,Z2 v(z^) z3 € Z x3€X z3 € ZH vCx^) adjacent 4 Lemma 5.2*, Let S* be an indecomposable star-closed set of vectors of length \/2 in Rn , with inner products +1, 0, or -1. Define a, b, X, Y, Z, and T as in Section 2. Then the inner products of x-, t ., z., x., y., z. are X X X J J J either as shown in Fig. 5*2, or as shown in Fig. 5*3* Proof. By Lemma 2.2, the inner product of two vectors x. and x- in X (orthogonal to a, inner product +1 with b) is either 0 or 1. Equations (2.1) - (2.4) imply that (5*D = a-b+x± , (5-2) zA = b-xi . If (xi,x-) = +1 , then the inner products are given by X J . Figure 2.6, reproduced as Fig. 5*2. On the other hand, if (x.,x.) - 0 , then the inner products are given by X J Figure 2.7i reproduced as Fig. 5*3* t------1 N / x , N t------1^ X . Fig. 5 .2. Type 1 connection, with (x.,x.) = 1 J Fig. 5*3. Type 0 connection, with (x^x^) J 95 The adjacencies in the base graph B(H*ab) defined in rules (1) - (6) correspond to adjacencies in the base graph B(S*:ab). Lemma 5»3« Let S* be a star-closed set of vectors. Let H be the graph of inner products of a subset S with no negative inner products. Let r^ be x-, y., or z., and let •L X JL <1 s - be x*, y., or z.. The vertices v(r•) and v(s-) are O t) J O ^ J adjacent in B(H:ab), if and only if the vectors x. and x. 1 J have inner product +1 in the subset X of S*. Proof. By Lemma 5-2, two vectors x. and x. in X of S# have 1 J inner product either +1 or 0. In the graph of inner products, two vectors with inner product +1 are adjacenti two vectors with inner product 0 are non-adjacent. There are two cases. Tvne I t If (x.,x.) = +1 . As shown in Fig. 5 .2, we have (x.,x.) = (y*,y.) - (z.,z.) - +1 , and J J A J (Xf.yj) - = = 0 . By rules (1) - (3) in B(H:ab), v(x-) is adjacent to v(x.), v(y.) is adjacent to A J A v(yj), and v(zi) is adjacent to v(zj). By rule (A), v(xi ) is adjacent to v(yj). By rule (5)» is adjacent to v(z^). By rule (6), v(y^) is adjacent to v(zj). Thus for Type 1, v(r^) and v(s-) are adjacent in B(Htab), where r* is x., y •, or z-, and s. is x., y., or z.. j J J j Type^Ji* If = ® ■ As shown in Fig. 5«3i we have ^xi ,xj^ ^ = ® • (Xf.y^) = —1 , and 96 (x-,z.) = (y.,z.) = +1 . By rules (1) - (3) in B(H:ab), J- J i J v(x.) is nonadjacent to v(x-), v(y.) is nonadjacent to v(y.), and v(z.) is nonadjacent to v(z.). Since (x. ,y.) J J 1 0 = -1 , x. and y. cannot both be in S. By rule (5), v(x-) -L J A is nonadjacent to v(z-). By rule (6), v(y.) is nonadjacent J **■ to v(z.). Thus for Type 0, v(r.) and v(s.) are adjacent v J in B(Hsab), where r. is x - , y . , or z. , and s. is x., y. t -1- -*■ J J o or z-. This completes the proof. J Let x., y ., z- be three vectors related by equations X -L iL (5*1) and (5*2). Since (x^z^) = -1 , a set S with no negative inner products cannot contain both x- and z-. J* J* Since (y^»z^) - -1 , a set S with non-negative inner products cannot contain both y^ and z^. However, S may contain both x- and y .. •L X Lemma 5.k*. Let H be a graph with an edge ab. Assume that none of Q, K2,3* or K5~LI 2111 induced subgraph of H. If x^ in XH and y^ in Y^ are adjacent in H, then no vertex x., y-, or z- has Type 0 connection to x^ and Type 1 J J J A connection to y^. Proof. If x. had Type 0 connection to x^ and Type 1 J J* connection to y^, then H would contain Q, as shown in Fig. If y. on Z j had Type 0 connection to Xi and J J Type 1 connection to y^, then H would contain K^-LI, as shown in Fig. 97 # a — * b X;x. * b Fig. 5.4. H does not contain Q or K^-LI Theorem 5.5. Let H be a graph, with an edge ab. If the least eigenvalue of H is ^ -2, then the base graph B(Hsab) is an induced subgraph of either CPCnOi^jK^ or the Schiafli graph. Proof. The least eigenvalue of H is ^ -2, so by Theorem 4.2, H is the graph of inner products of a set S of vectors in a root system +S*, that is or Eg. By Lemma 4.5» none of the graphs Q, Kg or K^-LI is an induced subgraph of H. The base graph B(H:ab) is defined by rules (1) - (6). Each vertex of V corresponds to a vector x^ in X of S*, as shown in Table 5»i» That is, if x^ is in X^, then v(x^) corresponds to the vector x^. If y^ is in yh' * then 'v(yj) corresponds to the vector x. = b-a+y- in S*. If z, is J 1 K in ZH , then vCz^) corresponds to the vector x^ = in S*. By Lemma 5* 3» a vertex v(x^) , or v(z^) is adjacent in B(H*ab) to a vertex v(x.), v(y-), or v(z.), u J J if and only if x^ and x^ are adjacent in B(S*iab). By Lemma if x^ in Xjj and y^ in y h are adjacent in H, then their adjacencies with any other x^, y^, or z a are J J J consistent. 98 Theorem 5.6. Let H be a graph, with an edge ab. Assume that each vertex of H is adjacent to a or b (or both), and that none of Q, 3 ’ or K^-LI is an induced subgraph of H. If the base graph B(Htab) is an induced subgraph of either CP(m)UK^ or the Schl&fli graph, then the least eigenvalue of H is > -2. Proof. First, assume that B(H:ab) is an induced subgraph of CPChOU k ^. Choose vectors a and b in Dm+3* such that (a,b) = +1 . Let X be the set of vertices of that are orthogonal to a and not orthogonal to b. Without loss of generality, (x,bl - +1 for any x in X. The base graph B(Dm+^sab) is the graph of inner products of X. By Theorem k.l, B(Dm+^:ab) is isomorphic to CP(m)[jK^. Thus there is an injective mapping 0, from the set V of vertices of B(H:ab) into the set X of vertices of B(Dm+^:ab), such that 0 preserves adjacency. Define a mapping ^from the set of vertices of H into the set of vectors of m+3 as follows. If x^ is in XH , then j^Cx^) - ©(vCx^). If in YH is adjacent to x^, then let /'(y^) = a-b+0(v(x^)) . If yj is in YH !,then let = a-b+0(v(y..)) . If is in ZH , then let = b-0(v(z^)). By Lemmas $.2 and 5*3» two vertices are adjacent in H if and only if their images under have inner product +1. Thus H is the graph of inner products of a subset of D m+3 By Theorem ^.2, the least eigenvalue of H is ^ -2 . On the other hand, assume that B(Hsah) is an induced subgraph of the Schl&fli graph. By a similar argument, there is an injective mapping YJ from the vertex set of H into the vector set of Eg, such that two vertices are adjacent in H if and only if their images under J^have inner product +1. Thus H is the graph of inner products of a subset of By Theorem 4.2, the least eigenvalue CHAPTER V CONSTRUCTION OF EXCEPTIONAL GRAPHS 1 * Introduction A graph is exceptional if it has least eigenvalue greater than or equal to -2, but it is not a generalized line graph. An exceptional graphs is represented as the graph of inner products of a set of vectors in the Lie algebra root system Eg. In this chapter, we use the base graph of Chapter IV to construct new exceptional graphs. Section 2 gives the steps in the construction of a graph H that is represented by a subset S of a root system S*. An edge ab in H corresponds to two vectors in S, with inner product (a,b) = +1. Any other vertex of H is in a subset Xjj, Yjj, Zjj, or Tjj, according as it is adjacent to b and not a, a and not b, both a and b, or neither a nor b, respectively. In the base graph B(S**ab), a symbol X, Y, or Z, represents a vertex in X^, Y^, or Z^. A symbol W represents two adjacent vertices, x^ in XH and y^ in Yjj. A sign (+ or -) represents a vertex in Tjj. In Section 3, the base graph construction is applied to construct exceptional trees. We show that there are just two exceptional trees represented in the root system E 100 101 Section 4 gives details of* the construction of* graphs represented in the root system E^. Table 4.1 presents the 42 different patterns of X, Y, and W that may occur in the base graph B* = L(K« „) . A particular X, Y p W pattern j i j restricts the possible patterns of Z and of T. In Table 4.2, we show that the number of exceptional graphs • represented in E^ is less than 30*000. Section 5 summarizes the study of exceptional graphs. We show the twenty smallest exceptional graphs (on six vertices). 102 2. Construction from a base graph By the following procedure, we construct any connected graph H that is represented in the root system Eg. Assume that H has an edge ah, and no induced subgraph Q, K0 or K<--LI. The vertices a and b are represented by two vectors a and b, with inner product (a,b) = +1 . The base graph B(Hiab) must be an induced subgraph of the base graph B* of a root system S*. For a graph H represented in the root system Eg, B* is the Schl&fli graph. In E^, B* is L(Kg); in Eg, B* is L(K^ ^). We call the vertices of B* cells. Mark some of the cells of B* with a symbol» X, Y, or W. A cell may have at most one symbol. The symbol X in cell i of B* indicates that the vector x^ in the root system S* represents a vertex x. in Xu , and that x- is i n i not adjacent to any y^ in The symbol Y in cell i of B* indicates that the vector y. in S* represents a vertex J* ^i and that y^ is not adjacent to any x^ in X^. The symbol W in cell i of B* indicates that the two vertices x^ and y^^ in S*, with (xi,yi) = +1 , represent two adjacent vertices, x^ in XH and y^ in YH> Two nonadjacent cells of B* cannot be marked X and Y, since in the edge-signed graph of Type 0 connection (Fig. IV.5*3)» and y^ are joined by a negative edge. For the same reason, two nonadjacent cells of B* cannot be marked X and W, or Y and W. 103 Mark other cells of* B* with a symbol * Z. The symbol Z in cell j of B* indicates that the vertex z. in the root J system S* represents a vertex in Z^. There are no restrictions on the adjacencies of cells marked Z. A cell not marked with a symbol X, Y, W, or Z, is "blank," and is not part of the base graph B(Hiab). The set T of vertices nonadjacent to both a and b is represented in the root system S*. by the set T* of vectors orthogonal to the vectors a and b. Mark each vector of T* with a sign* + or - or 0. A + sign on t. . - 2t.-x. J * J indicates that the vector t.. in T* represents a vertex X J in T„j A -.sign on t* . indicates that the vector -t. . n. ij represents a vertex in T^. The next four paragraphs give the restrictions on the possible signs in T*. For each vertex t^.. in T*( there are 3 (in E^), 4 (in E ^ ) , or 6 (in Eg) vertices k of B*, such that the inner product (xk*'tij) = 1 • If any of these vertices of B* is marked X, Y, or W, then the sign on t. . must not be If any.of these vertices of. B# is marked Z, then the sign on t^j must not be +. For each vertex t - . in T*( there are 3, 4, or 6 J vertices k of B* such that the inner product (x,_,t. .) = -1 . •K 1 J If any of these vertices of B* is marked X, Y, or W, then the sign on t •- must not be +. If any of these vertices in B* is marked Z, then the sign on t .. must not be 10^ Each vertex of T* can be written in 3, 4, or 6 ways as the difference of two vectors in X, t.. = x.-x. . j* J x . 3 The sign ©4 on and the sign £ on t^, must be chosen so that H has no negative edges. That is, (eit.1 J•,£t,. xliC ) = = ^ ( ( x i ,xh )-(xi ,xk )-(x;.,xh )+(xj ,xlc)) must be either +1 or 0 , not -1. Since H is to be connected, each vertex in T* with nonzero sign must be connected (by a path of nonzero signs in T*) to a non-blank vertex of B*. These symbols and signs.determine a connected graph H, represented in the root system. The least eigenvalue of H is greater than or equal to -2. If H is not a generalized line graph, then H is an exceptional graph. .105 3. Exceptional trees An exceptional tree is a connected graph, with no cycles, that has least eigenvalue greater than or equal to -2 , and is not a generalized line graph. In this section, let H he an exceptional tree. Thus H is represented in the root system Eg. The hase graph of H is an induced subgraph of the Schl&fli graph. ' As Fig. 3*1 shows, none of the cells of B* may be marked with the symbol Z, since a b z a would represent a 3-cycle in H. None may be marked W, since a b w^ a would represent a ^-cycle a b x. y. a in H. No two cells J, X marked X are adjacent, since b x x' b would represent a 3-cycle in H. Similarly, no two cells marked y are adjacent since a y y* a would represent a 3-cycle in H. By the rules of Section 2, each cell marked X must be adjacent to each cell marked Y. Since the corresponding x and y are adjacent in the base graph B(Hxab), they represent nonadjacent vertices in H. a i b x i Fig. 3*1• Cycles in H are not permitted 10 6 These restrictions are satisfied by only six different patterns in the Sch.lS.fli graph. The six patterns are shown in Fig. 3*2. Fig. 3.2. The six base graphs of exceptional trees We now consider in detail the exceptional trees that are represented in E^. A vertex nonadjacent to a and b in H is represented by a vector t. . = x .-Xj in T*. X J The six T* vertices of are shown in Fig. 3.3. Each one can be written in three ways, for example: t^2 = t ^ = t^Q , and = t25 = tllf . 107 Fig. 3.3. The six T vertices in E The base graph of Fig. 3*2 is represented in Eg by an X in cell 1 of B* - L(K^ . The sign on t12 is not - since the inner product = (x^.-x^+Xg) « -2 + 1 is negative. The possible sighs are 1 + or 0 on t^gj + or 0 on t ^ j +, or 0 on tg^* + or 0 on t^gj + or 0 on ^39* +' or 0 on ^69* We possible signs for the six T vectors in B^, in that order* (+ + ± » + t ± ) • Since 0 is always possible, we do not write 0, unless it is the only possibility. We will show that the two trees in Fig. 3.^ are the only exceptional trees that are represented in Eg. T—* Fig. 3.^. The two exceptional trees in Eg 108 Theorem 3.1. The only exceptional trees that are' represented in Eg are the two trees of Fig. 3• ^•• Proof. We construct a tree H represented in E^. The line graph B(Hiah) is contained in B* - L(KQ ~) . The six possible choices of symbols X and Y are shown in Fig. 3.2. The base graph of a tree cannot have any symbols W or Z, as shown in Fig. 3.1. It remains only to choose signs in T*. For the base graph B1 of Fig. 3*2, represented by x^ only, the signs for It^2 ^23 * ^36 ^39 ^69^ are c*losen from the possibilities (+ + + , + + + ) , respectively. Recall that 0 is always permitted for any or all of T*. There are only five different choices, shown in Fig. 3*5* The signs (0 0 0 , 0 0 0) construct H - P^ , which is a line graph (of P ^ ) , hence not an exceptional graph. The signs (0 + p , 000) construct H = P^ = Xj(P^), also not an exceptional graph. The signs (0+0 , 0+0) construct H - L(P^j1,0,0,0), a generalized line graph, hence not an . exceptional graph. Finally, the signs (0+0 , 0 + +) and (0 + + , 0 + +) construct the two exceptional trees of Fig. 3*^* Now consider the base graph pattern Bg of Fig. 3*2, represented in L(K^^) by nonadjacent x1 and x^. Neither +t12 nor -t12 is allowed, since ( ^ , - t ^ ) =. (x1,-x1+x2-) - -1 , and (x^,t12) = (x^,t^) = (x^,x^-x,j) = -1. Thus the possible signs are (0++,0++). As Fig. 3*6 shows, the signs 1 0 9 (000 , 0 0 0) and (0 + 0 , 0 0 0 ) construct generalized line graphs MP^; 1,0,0) and L(P^;1,0,0,0), respectively. The signs (0 + + , 000) construct a five-cycles b Xl t ^ t23 x5 b , which is forbidden in a tree. The signs (0 + 0 , 0 + 0) construct L f P y 1,0,1). The signs (0+0 , 0 0 +) construct the first exceptional tree of Fig. 3.2. Base graphs B^, B^, B^, and B^ are treated similarly, and produce no other exceptional trees. The possible signs for B^ are (0 0 0 , 0 0 0), constructing L(P2*1,1). The signs for B^ are (0 + + , 0 0 0), for B^ are (0 0 + , 0 0 +), and for Bg are (000 , 0 0 0 ). This completes the proof. ( 0 0 0 , 0 0 0 ) (0+0,000) (0 + 0 , 0 + a) a f a '13 at tl3t bo -*t x. 39 LCPj*) l (p 5 ) L(Pj^;1,0,0,0) (0 + 0 , 0 ++) (0++ , 0++) ►t. “r *13 '69 bt3 tl3I_C94 * t b.--- P 3 I X. X, 39 Exceptional tree Exceptional tree Fig. 3*5* Signs for T in B^, and the graph H 110 (000,000) (0+0,000) (0 + + , 0 0 0) a af t. a T ' *131 I « x< b i Xx bt------#x, CL- x 13 23 l (p 3 *i ,o ,o ) L(PJifil,0,0,0) 5-cycle (0 + 0 , 0 + 0) (0 + 0 , 0 0 +) a a T T t13 13 b »t b 39 xi 69 L(P3il,0,l) Exceptional tree Fig. 3.6. Signs for T in B„, and the graph H Ill 4. Graphs represented in E^ The base graph B* = L(K^ of , and the six associated sectors of T*, are shown in Fig. 3*3* If one cell of B* is marked X, Y, or W, and another cell in the same line is marked 2, then the two T*-vectors joined to that line must have 0 signs. For example, with cell 1 marked Z and cell 4 marked X, (+‘fci2*zl^ = ' anc* (-t^2 ,x^) = -1 , so the sign on t^2 must be 0. At most two of the three edges among —^13* — ^23 are positive, so at most two of the three signs can be nonzero in the construction of a graph H. When constructing a connected graph H, the sign (if nonzero) of t . . is limited to at most one of + or to avoid negative inner products with a cell of B* or another vertex of T*. The number of combinations of t^2, 0, 1, 2, or 3 of the three have nonzero sign, is 1+3+3+O = 7 . Similarly, the number of combinations of ^3 6 ’*3 9 ' ^69 in which 0 , 1 , 2 , or 3 of the three have nonzero sign, is also 7 . Since the first three vectors are orthogonal to the other three, the choices are independent. Thus, for a given pattern of X, Y, W, Z in L(K^ ^), there are at most 49 choices of signs in T* that produce connected graphs H . 112 Table 4.1 shows the 42 possible patterns of X, Y, W in the base graph of E^. These are arranged in three classes. Class 1 (patterns 1 throhgh 2 6 ) have only X symbols. If a pattern has only Y symbols, then without loss of generality, we may reverse a and b, to change every Y to X. Class 2 (patterns 27 through 32) have some cells marked X t and others marked Y. Because of the inner products of vectors in a root system, each cell marked X must be adjacent in B* to each cell marked Y. Class 3 (patterns 27 through 32) has cells marked X, Y, and W. Because of the inner products of vectors in a root system, each cell marked W must be adjacent to each cell marked X, Y, or W. Table 4.1 also shows the signs on T that are possible for a given pattern of X, Y, W. The sign 0 is always allowed, so it is not written in the table (unless it is the only possibility). If i of the 9 cells of B* « L(1C -) are marked X, Y, J i .3 or W, then any or all of the remaining 9-i cells of B* may 9—i be marked with the symbol Z. Thus, there are at most 27 possible combinations of Z-cells. This appears in the last column of Table 4.1. 113 TABLE *K1 THE hZ PATTERNS OP X, Y, W IN E^ Pattern # Base graph Possible T Possible Z (Class 1. X only, 26 patterns) i jzf i l l > ^ ^ i 2^ + + +,+ + + 28 X 0 + + , 0 + + 27 X X X 0 + + , + + + 27 5 x 000,000 2 X X 6 X X 000,0 + + 26 X 7 XX 0 + +.0 + + 2^ X 8 XXX 000,+++ 26 9 XX 000,000 25 X X 10 XX 000,000 2-5 X X 114 TABLE 4.1— Continued Pattern # Base graph Possible T Possible Z 11 X XX 0 0 0 , 0 + t 25 X X 12 X X X 000,0 + + 25 X 13 XX 0 + + , 0 + + 25 X X 14 XX 000,000 2^ X X X 15 XX 0 0 0 , 0 0 0 2^ X X X 16 XXX 000,000 2^ X X 17 XXX 000,000 2^ X X 18 XXX 000,0 + + 2^ X X 19 XX 0 0 0,000 zJ X X X X 20 XXX 000,000 23 x x X 21 XXX 000,000 2^ X X X \ 115 TABLE 4-.1— Continued Pattern # Base graph Possible T Possible Z 22 XXX 0 0 0 , 0 + + *3 XXX o o o 23 X X X m 0 0 0 22 X X XX 2*f X XX 0 0 0 , 0 0 0 22 X XX X 25 XXX ooo, 0 0 0 21 X XX XX o o o 26 XX X w 0 0 0 2° XXX XX X (Class 2* X and Y, 6 patterns) 27 Y X 0 + + , + + + 28 YXX 000,+ + + 26 29 YX 0 + + , 0 + + 26 X 30 YXX 000,0 + + 2^ X 31 YX 0 + +,0 + + 25 X Y 32 Y X X 000,000 A X X 116 TABLE 4.1— Continued Pattern # Base graph Possible T Possible Z (Class 3* X, Y, and W, 10 patterns) 33 W + + +. i + + i 34 W W 0 + + , + + + 2? 35 W W W 0 0 0,+++ 26 3 6 WX 0 + +,+ + + 37 WXX 0 0 0,+ + + 26 38 WX 0 + + , 0 + + 26 X 39 WXX 0 0 0,0 + + 2? X 40 WXX 000,000 2** X X » 41 X W W 0 0 0,+ + + 26 42 XYW 000,+++ 26 117 In pattern 2, the possible signs for vectors t12 t ^ t«« are + + +. The number of possible combinations with 0, 1, 2, or 3 of t12 t ^ t2j nonzero is* 1+3+3+0 = 7 . In pattern 3, with signs 0 + +, the number of possible combinations with these three vectors isl+2+l+0=5« In pattern 5* with signs 0 0 0, the number.of possible combinations with 0, 1, 2, or 3 of these three vectors nonzero is* 1+0+0+0 = 1 . These estimates are used in Table if.2. Theorem if.l. There are fewer than 30*000 connected exceptional graphs represented in the root system E^. Proof. A graph H represented in E^ is constructed by marking up to nine symbols X, Y, W, Z on the base graph B* - L(K_ «), 3*3 and six signs (+, -, or 0) on the vectors of T*. In Table if.2, for each one of the if2 .patterns of X, Y, W, there are upper bounds on the number of possible .choices oft signs on t12 t ^ t2^, signs on t^g t ^ t^» and symbols Z. The product of these upper bounds is dn upper bound on the number of graphs H constructed with that pattern of X, Y, W. Without loss of generality, we make the following reductions for patterns 1, 2, and 33* which would otherwise be qi^ite large. Consider pattern 1, which has no symbols X, Y, W. 118 If two nonadjacent cells in B* are marked Z, representing adjacent vertices Zg and z^ in H, then the same graph H is generated by a base graph BCHsZg b ) , as shown in Fig. 4.1. If there is a vertex in T, joined to a cell marked Z, then the same graph H is generated by a base graph B(H*a z), as shown in Fig. 4.1. If all the signs on T are zero, and there are no nonadjacent cells of B* marked Z, then the graph H is for some m, a line graph, hence not an exceptional graph. Therefore, pattern 1 need not be counted at all in Table 4.2. Now consider pattern 2, which has one cell marked X. There are only 49 different patterns of Z in the presence of one x. The..number of patterns - with 0, 1, 2, ..., 8, Z's 1st.1« 2, 6, 10, 11, 10, 6, 2, 1, respectively. Q Thus 49 replaces 2 as an upper bound on Z in pattern 2. The same is true in pattern 33, which consists of a single W. The subtotals of upper bounds, for the patterns in Classes 1, 2, and 3 are 10,864 and 5.712 and 12,592, respectively. In total, there are at most 29,105 connected exceptional graphs represented in the root system Eg. This completes the proof. a z a a Fig. 4.1. Substitutions'for Pattern 1 119 TABLE 4.2 AN UPPER BOUND ON THE NUMBER OP EXCEPTIONAL GRAPHS IN E, Pattern # Possible T Possible Z Uoner bound (Class 1* X only, 26 patterns) 1 --— 0* 2 7 7 49* 2,401 3 4 4 27 2,048 4 4 7 27s 3,584 5 1 1 2 64 6 1 4 26> 256 7 4 2 1,024 8 1 7 26 448 9 1 1 23 32 10 1 1 2 32 11 1 4 25 128 12 1 4 25 128 13 4 2^1, 512 14 1 1 2 | „ 16 15 1 1 16 24 16 1 1 2 I. 16 17 1 1 2 16 18 1 24 64 19 1 23 8 20 1 23 8 21 1 1 2? 8 22 1 23 32 23 1 1 2? 4 24 1 1 2 4 25 1 1 21 2 26 1 1 2° 1 Subtotal 10,864 120 TABLE 4.2— Continued Pattern # Possible T Possible Z Upper bound (Class 2. and Y, 6 patterns) 2? 4 7 3,584 28 1 7 448 29 4 4 1,024 30 1 4 128 31 4 4 512 32 1 1 16 Subtotal 5,712 (Class 3. X, Y, and W ( 10 patterns) 33 7 7 49* 2,401 34 4 7 3.584 35 1 7 448 36 4 7 3.584 37 1 7 448 38 4 4 1,024 39 1 4 128 40 1 1 16 41 1 7 448 42 1 7 448 Subtotal "127592 Total 29,105 * Reduction, described in text. 121 This upper bound is greater than the number of exceptional graphs represented in E^, for several reasons. A cell marked Z may rule out a sign for a vector of T. 9—i Some of the 2r combinations of Z are symmetric to others. There is symmetry, in some cases, between t^2 t ^ ^23 an<* ^36 ^39 ^69 ' constructed from two base graph markings may be isomorphic (for different choice of ab). Finally, a graph that is represented in may be a generalized line graph, hence not exceptional. Some exceptional graphs, and their base graph diagrams, are shown in Appendices B and C. r 122 5. Exceptional graphs A graph has least eigenvalue greater than or equal to -2, if and only if it is the graph of inner products of a set of vectors in a root system Dm or Eg. Generalized line graphs are represented in Em . A graph is exceptional if its least eigenvalue is ^ -2, and it is not a general ized line graph. A graph represented in the root system Eg is constructed with the base graph B* « L(K^ , which has 9 cells. We showed in Section k that there are k2 patterns of symbols Q—i X, Y, and W, filling i cells* 27 combinations of Z-cells* and at most *+9 combinations of the six vectors in T. As Table k.2 shows, the number of connected exceptional graphs in Eg is less than 30,000. A graph represented in the root system E^ is constructed with the base graph B* = L(Kg), which has 15 cells, A pattern of symbols X, Y, W, Z, or blank fills the cells of B*. There are 15 vectors in T. In the construction of a connected exceptional graph, after the set of nonzero T vectors is chosen, their signs are determined. Thus the number of connected graphs represented in E? is less than 515 215 = 1015. 123 A graph that is represented, in the root system Eg is constructed with the base graph B* isomorphic to the Schl&fli graph, which has 27 vertices. A pattern of symbols X, Y, W, 2, or blank fills the cells of B*. There are 36 T vectors. In the construction of a connected exceptional graph, after the set of nonzero vectors is chosen, the signs are determined. Thus the number of connected exceptional graphs is less than 52^ 2-^ = 512 x 1027 . The eigenvalues of the 1^3 connected graphs on 1 to 6 vertices are tabulated in Appendix A. Twenty of these have least eigenvalue greater than or equal to -2, and are not generalized line graphs (by inspection). These 20 smallest exceptional graphs are shown in Fig. 5*1« All are represented in Eg, as shown by their base graph diagrams. 124 - o o <^> ^ 5 6 7 8 U < Z > 9 10 11 12 13 15 16 17 18 19 20 Fig. 5.1. The twenty smallest exceptional graphs CHAPTER VI THE LINE MUI/EIGRAPH OP A MULTIGRAPH 1. Introduction A multigraoh is a finite nonempty set of vertices, and a family of edges. Recall that in a graph, two vertices are joined by at most one edge. In a multigraph, two vertices may be joined by t edges, for any non-negative integer t. However, an edge must still have two distinct end-vertices. This chapter is arranged as follows. Multigraphs are defined in Section 2. The line multigraph of a multigraph is defined in Section 3 . We introduce three important multigraphs that are not induced sub-multigraphs of a line multigraph. In Section 4, we prove geometric characterizations of line multigraphs. These theorems differ only slightly from the geometric characterizations of line graphs, in Section 2 of Chapter I. In Section 5* we characterize the multigraphs that have least eigenvalue greater than or equal to -2. The theory of Chapter III, concerning graphs with this property, is applied to multigraphs. 125 The final result of this dissertation is Theorem 5-3* A multigraph M has least eigenvalue greater than or equal to -2, if and only if it is the multigraph of inner products of a family of vectors (with non-negative inner products) in the root system Dn or the root system Eg. 127 2. Multigraphs Let V be a set of vertices. An edge vw is an unordered pair of distinct vertices. The ends of the edge are the two vertices v and w. Two edges vw and xy are equal, if v = x and w = y , or v = y and w = x . A family F of edges is a list of edges. Various edges of the list may be equal to each other. If an edge is equal to t-1 other edges in F, then the multiplicity of the edge in the family F is t. A multigraph M - (V,F) is a finite nonempty set V of vertices, together with a family F of edges. Multiple edges are permitted. (Several edges may have the same ends.) Loops are not permitted. (The two ends of an edge must be distinct vertices.) A t-bond is a multigraph on two vertices, with t edges (t £ 1) joining them. Two vertices of a t-bond are t-ad.iacent. Two vertices with no edge joining them are non-ad.iacent. The degree of a vertex in a multigraph M = (V,F) is the number of edges in the family F that have that vertex as an end. A multigraph is regular if every vertex has the same degree. For example, a t-bond is a multigraph on two vertices, regular of degree t. 128 Lemma 2.1. In a multigraph, the number of edges is equal to half the sum of degrees. Proof. Each edge in the family contributes 2 to the sum of degrees. A sub-family of a family F, is a family F*, such that each edge in F* is in F, with multiplicity in F greater than or equal to its multiplicity in F'. A sub-multigraph of a multigraph M = (V,F) is a multigraph M' - (V',F') , such that V' is a subset of V, and F' is a sub-family of F. An induced sub-multigraph of M is a multigraph M" = (V",F") , such that V" is a subset of V, and F" is the family of all edges in F that have both ends in V". The underlying set u(F) of a family F, is the set such that each edge in the family F is equal to exactly one edge in the underlying set u(F). The underlying graph of a multigraph M = (V,F) is the graph G =* (V,u(F)) . Thus, the underlying graph G has one edge for each t-bond (t £ 1) in M. Let M = (V,F) and M' = (V*,F*) be multigraphs, a multigranh isomorphism is a bisection T i V —>V* , such that for any two vertices i and j in V, the number of edges in F that join i and j is equal to the number of edges in F' that join T(i) and T(j). 129 3. The line multigraph Let M = (VfF) be a multigraph. Two edges of M have either 0, 1, or 2 ends in common. The line multigraph L(M) is a multigraph whose vertex set corresponds to the family F of edges of M. Two vertices of L(M) are joined by t = 0, 1, or 2 edges, if in M they correspond to two edges with t ends in common. Accordingly, two vertices of a line multigraph are either non-adjacent, 1-adjacent, or 2-adjacent. Lemma 3.1. If a multigraph JVI has p vertices, q edges, and vertex degrees d^, dg» dp, then the line multigraph L(M) has q vertices and q^ edges, where* Proof. A vertex of degree d^ in M corresponds to id^(d^-l) edges in L(M). Thus the number of edges of L(M) is equal to X d^l) = isOLd-2 - i Z l d i . By Lemma 2.1, the number q of edges of M is equal to half the sum of degrees of M. Let (e^.... e^) be the edges of a t-bond in M, with ends x and y. Let f be an edge with x as one. end.. Let g be an edge with neither x nor y as ends. In the line multigraph L(M), the vertices e^, e-t are 2-adjacent to each other, 1-adjacent to f, and non-adjacent to g. 130 We define a t-cluster in a multigraph N, to "be an induced sub-multigraph consisting of t vertices, that are all 2-adjacent to each other, all 1-adjacent to a fixed neighborhood, and all non-adjacent to the rest of the vertices of N. Thus a t-bond in M corresponds to a t-cluster in the line multigraph N = L(M)’. -This is illustrated.in Fig, 3.1, f Fig. 3.1. A 4-bond in M, and the 4-cluster in L(M) Lemma 3 .2 . Let N be a multigraph, with no t-bonds for t > 2 . Each 2-bond of N is in a cluster, if and only if none of the three multigraphs of Fig. 3.2 is an induced sub-multigraph of N. Proof. Let x and y be 2-adjacent in N.' The number of edges from a vertex z to (x,y) is either* (0 ,0 ), (1 ,1), (2 ,2 ), (0 ,1 ), (0 ,2 ), (1 ,0 ), (1 ,2 ), (2 .0 ), or (2 ,1). In a cluster, only the first three are allowed. The other six possibilities induce the multigraphs of Fig. 3 .2 . Fig. 3.2. The three forbidden multigraphs 131 Let N be a multigraph with no t-bond for t > 2 , such that each 2-bond is in a cluster. Notice that a cluster is maximal* since the vertices of the cluster are all 1-adjacent to a fixed neighborhood, and all non- ad jacent to the other vertices of N. A 1-cluster is a vertex that is not an end of any 2-bond. Let C be a set of vertices of N, with exactly one vertex in C from each cluster of N. The cluster-reduction graph H of N is the sub-multigraph of N that is induced by the vertex set C. Lemma 3.3. The cluster-reduction graph H is a graph. Proof. By assumption, N has no t-bond for t > 2 . Each 2-bond of H is in a cluster. By definition of G, two vertices in C are not in the same cluster of N. Thus, they are either 1-adjacent or non-adjacent. Define the multiplicity function m * C —>{l,2,...} , where m(c) is the number of vertices in the cluster of N that contains the vertex c. Thus c in H represents an m(c)-cluster of N. 132 Recall from Chapter I that is the line graph of both and y but no other graph is the line graph of more than one graph. Lemma 1.2.3. Let G and G' be connected graphs with isomorphic line graphs. Then either G and G' are isomorphic,graphs, or else they are and y We prove an analogous result for multigraphs. Lemma 3.^-. Let M and M' be connected multigraphs with isomorphic line multigraphs. Then either M and M' are isomorphic multigraphs, or else their underlying graphs are among the graphs of Fig. 3-3* Proof. The line multigraph L(M) has no t-bond (t > 2), and each 2-bond of L(M) is in a cluster. Thus L(M) has a cluster-reduction graph H. Let G be the underlying graph of M. Then H - L(G) . Similarly, H' ~ L(G*) , where H' is the cluster-reduction graph of L(M*)» and G* is the underlying graph of M*. The isomorphism from L(M) onto L(M') restricts to an isomorphism from H to H 1. By Lemma 1.2.3, if G and G* are not isomorphic, then they are and first two graphs in Fig. 3 .3 . 133 If the isomorphism from H to H* induces an isomorphism from G to G*, then a bond of M and the corresponding bond of M* have the same number of edges* hence M and M' are isomorphic multigraphs. However, if the isomorphism from H onto H* does not induce an isomorphism from G onto G*, then H contains two even triangles with a common edge, corresponding to and in G and to ^ and in G*. Since M is connected, H is connected. Since H is a line graph, it has no induced subgraph Since the two triangles are even, no other vertex of H is adjacent to an oddnnumber of the three vertices in a triangle. The only graphs H with these properties are the three graphs of Fig. 3 .^. Their line graphs are the last three graphs of Fig. 3 .3 , Fig* 3*3* Five underlying graphs G, G* Fig. 3*^* Three cluster-reduction graphs H, H* 13^ Fig. 3*5 shows four examples in which nonisomorphic multigraphs M and M' have isomorphic line multigraphs L(M) = L(M'). M '-m \ M 1 L(M) =L(M*) Fig* 3*5* Multigraphs and line multigraphs 135 4. Geometric characterizations The line graph of a graph is characterized by the following theorem from Chapter I. Recall that a triangle of three adjacent vertices in a graph H is called odd if some vertex of H is adjacent to an odd number of the three vertices. Theorem 1.2.4. The following are equivalent. (1) H is a line graph. (2) The edges of H can be partitioned into complete subgraphs, such that no vertex is in more than two of the subgraphs. (3) Ki 3 Is no,t an induced subgraph of H, and if two odd triangles have a common edge, then the subgraph induced by their vertices is K^. (4-) None of the nine graphs G^^ - G^ of Fig. 4.1 is an induced subgraph of H. G i Y G^ g7 Fig. 4.1. The nine graphs forbidden in a line graph 136 For the analogous characterization of line multigraphs, we will need the following lemma. Lemma b .1. Let N he a multigraph with no t-hond for t > 2 , such that each 2-bond is in a cluster. If the cluster- reduction graph H is a line graph, then N is a line multigraph. Proof. Let C be a set of vertices of N, with exactly one vertex in C from each cluster of N. Then H is the induced sub-multigraph of N on the vertex set C. By hypothesis, H is the line graph of a graph G = (V,E) • A vertex c of H corresponds to an edge of G. Let M be a multigraph with vertex set V. If a vertex c in C is in a cluster of m(c) vertices in N, then let the corresponding edge of G be the underlying edge of an m(c)-bond of M. Then N is the line multigraph of M. A triangle in a multigraph, is an induced sub-multigraph on three vertices that are 1-adjacent to each other. A triangle in a multigraph N is odd if there is a vertex of N that is either 1-adjacent to all three vertices of the triangle, or else 1-adjacent to one of the three, and non-adjacent to the other two. The proof of the following theorem follows the analog ous proof by Harary (1969* PP* 7^-77) of the three character izations of line graphs. 137 Theorem 4-.2. Let N he a multigraph, with no t-bond for t > 2 , such that each 2-bond of N is in a cluster. The following are equivalent. (1') N is a line multigraph. (2*) The edges of N can be partitioned into cliques, such that no vertex of N is in more than two of the cliques. (31) Neither y nor with both triangles odd, is an induced sub-multigraph of N. (V) None of the nine graphs in Fig. 4-.1 is an induced sub-multigraph of N, Proof. (1*) implies (2*) Let N be the line multigraph of a multigraph M. A vertex i of M, with degree d^, is repre sented by a clique of d^ vertices in N. Let S be the set of those cliques such that d^ > 1 . (A clique with d^ = 1 has no edges.) This partitions the edges of N. Now a vertex of N represents an edge of M, which has two ends in M. Thus the vertex of N is in at most two of the cliques. (21) implies (1') Let S be a set of cliques partitioning the edges of N, such that no vertex of N is in more than two of the cliques.. Let U be the set of vertices of N that belong to only one of the cliques of S. Let M be the t intersection multigraph with vertex set S(JUj two vertices of M are t-adjacent if, as sets, their intersection contains t vertices of N. Then N is the line multigraph of M. 138 (21) implies (4*) If N has such a partition into cliques, then none of the nine forbidden subgraphs is an induced subgraph of N, since none of them has such a partition of edges into cliques. (4*) implies (V) Assume that N does not have property (3')» to show that N does not have property (V). If ^ is an induced subgraph of N, then so is G^ of Pig. **.1, since G^ = ^ . Let K^-K2, with both triangles odd, be an induced subgraph of N. Some vertex of N is 1-adjacent to an odd number of the three vertices of one triangle, and non-adjacent to the other vertices of that triangle. Also, some vertex of N (possibly the same vertex) is 1-adjacent to an odd number of the three vertices of the other triangle, and non-adjacent to the other vertices of that triangle. As in the case of graphs, the graph on K^-Kg with the other one or two vertices either contains ^ or else is one of the other eight graphs of Pig. 4.1. (3*) implies (11) Since N has no t-bond for t > 2 , and each 2-bond of N is in a cluster, N has a cluster-reduction graph H. There is one vertex of H for each cluster of N. Suppose that H is not a line graph. By Theorem 1.2.4, H does not have property (3)» i.e. H has either K- or t j K^-K2 with both triangles odd, as an induced subgraph. Then the same is true of N, contrary to (3'). Thus H is a line graph. By Lemma 4.1, N is a line multigraph. This completes the proof. 139 Theorem 4.3. A multigraph N is a line multigraph, if and only if: N has no t-bond for t > 2 , and none of the nine graphs of Fig. 4.1, nor the three multigraphs of Fig. 3»2, is an induced sub-multigraph of N. Proof. Let N be the line multigraph of a multigraph M. Then two vertices of N represent two edges of M, which have at most two ends in common. Hence N has no t-bond for t > 2 . Also, since N is a line graph, each 2-bond of N is in a cluster, representing a t-bond of M, By Lemma 3*2, none of the three multigraphs of Fig. 3*2 is an induced sub-raultigraph of N, By Theorem 4.2, none of the nine graphs of Fig. 4.1 is an induced sub-multigraph of N. Conversely, let N be a multigraph with no t-bond for t > 2 and none of the twelve sub-multigraphs. By Lemma 3*2* each 2-bond of N is in a cluster. By Theorem 4.2, N is a line multigraph. l k o 5. Multigranhs with least eigenvalue ^ -2 Let N "be a raultigraph on p vertices. The adjacency matrix of N is the p-square matrix A(N), with (i,j)-entry equal to t if vertex i and vertex j are t-adjacent in N. Thus A(N) is a real symmetric matrix, with 0 on the main diagonal, and entries 0 and 1 elsewhere. The eigenvalues of a multigra-ph are the eigenvalues of its adjacency matrix. The eigenvalues of any real symmetric matrix are real numbers. Thus the eigenvalues of a multigraph are real. Lemma 5.1. A t-bond for t > 2 , and the three forbidden multigraphs of Fig. 3*2, all have least eigenvalue less than -2 . - Proof. The eigenvalues of a t-bond are t and -t. The eigenvalues of are 0 and + \/5» which are approximately: 2.236, 0.000, and -2 .236. The eigenvalues of Mg are 0 and + \/Q , which.are‘approximately: 2 .828, 0 .000, and -2 .828. The eigenvalues of are 1 and IK-1+\/33)» which are approximately: 2.372, 1.000, and -3*372. 1 The next theorem reduces the characterization of multigraphs with least eigenvalue greater than or equal to -2 , to the characterization of graphs with least eigenvalue greater than or equal to -2. Ikl Theorem 5.2. A multigraph N has least eigenvalue greater than or equal to -2, if and. only if: (1) N has no t-bond for t > 2 , (2) None of the three multigraphs of Fig. 3*2 is an induced sub-multigraph of N, and (3) The cluster-reduction graph H of N has least eigenvalue greater than or equal to -2 . Proof. Let N be a multigraph with least eigenvalue greater than or equal to -2. By Cauchy's inequalities, the least eigenvalue of N is less than or equal to the least eigenvalue of any induced sub-multigraph. By Lemma 5»1» N has no t-bond for t > 2 , and none of the three multigraphs of Fig. 3*2 is an induced sub-multigraph of N. By Lemma 3 .2 , each 2-bond of N is in a cluster. Thus N has a cluster-reduction graph H. The eigenvalues of the matrix C* = A(N) + 21 are non-negative, if and only if C* is the Gram matrix of inner products of a family S* of vectors in Rn * Since the entries on the main diagonal of C' are 2, the vectors in S' have length \/2. Since the other entries of C* are 0 , 1, or 2 , the vectors have inner product 0, 1, or 2. That is, the vectors are at angles of 90°, 60°, or 0°. Let S be the underlying set of vectors in the family S'. Then the Gram matrix of S is A(G) + 21 . Therefore, the least eigenvalue of H is ^ -2. 1A2 Conversely, let N, be a multigraph, with properties (1), (2), and (3). The cluster-reduction graph H has least eigenvalue greater than or equal to -2 , so the eigenvalues of the matrix C = A(H) + 21 are non-negative. Thus C is the Gram matrix of a set S of vectors in Rn . The vectors in S have length \/Zt and distinct vectors have inner products 0 or 1. Let S' be the family of vectors such that each vector of S occurs in S' as many times as there are vertices of N in the cluster represented by the corresponding vertex of H. Then the Gram matrix of S' is A(N) + 21 . Therefore, the least eigenvalue of N is ^ -2. Let S' be a family of vectors of length y/z in Rn , with inner products -2, -1, 0, 1, or 2. The bond-signed multigranh of inner -products of S' is a multigraph with vertex family S ', together ;with a. sign on each bond, that indicates the inner product: -2 by a negative 2-bond, -1 by a negative 1-bond, 0 by no edge, +1 by a positive 1-bond, and +2 by a positive 2-bond. Theorem *5.3. A connected multigraph N has least eigenvalue greater than or equal to -2, if and only if it is the multigraph of inner products of a family S' of vectors in a root system Dn or Eg, such that S' has no negative inner products. Proof. Let N "be a connected multigraph with least eigenvalue ^ -2. By Theorem 5.2, the cluster-reduction graph H of N has least eigenvalue greater than or equal to -2. Since M is connected, H is connected. By Theorem k.2 of Chapter IV, H is the graph of inner products of a set S of vectors in a root system Dn or Eg. The set S has no negative inner products. Let S' he a family of vectors, such that each vector of S occurs as many times in S' as there are vertices of N in the cluster represented hy the corresponding vertex of G. Then N is the multigraph of inner products of the family S'. Conversely, let N he the multigraph of inner products of a family S' of vectors in a root system Dn or Eg. Let S he the underlying set of S'. Then the cluster-reduction graph H of N is the graph of inner products of the set S. Since M is connected, H is connected. By Theorem k.2 of Chapter IV, S is a subset of a root system Dn or Eg. Therefore, S* is a family of vectors in Dn or Eg. Since N has no negative bonds, S has no negative inner products. This completes the proof. This theorem generalizes the main theorem of Cameron et al. (1976), from graphs to multigraphs. v APPENDIX A THE 1*1*3 CONNECTED GRAPHS ON 1 TO 6 VERTICES How many graphs are there? If p vertices are labeled from 1 to p, then there are possible combinations of edges joining pairs of vertices. However, the number of isomorphism classes is less. Two graphs are isomorphic if there is a bisection from the vertex set of one graph onto the vertex set of the other graph, that preserves adjacency. The number of isomorphism classes of graphs on 1 to 6 vertices is shown in Table A.I. TABLE A . 1 THE NUMBER OF GRAPHS ON 1 TO 6 VERTICES Number of Labeled Isomorphism Connected Vertices Graphs Classes Graphs 1 1 1 1 2 2 2 1 3 8 4 2 4 32 11 6 5 1,024 34 21 6 32,768 156 112 Total 33.835 208. 143 SOURCE: Harary (1969, PP- 214-224). 144 1U-5 Representatives of the 208 isomorphism classes are drawn in Appendix 1 of Harary (1969)* To distinguish the non-isomorphic graphs with the same number p of vertices and Q of edges* Harary has numbered the graphs with the same P and Q. Table A .2 presents the eigenvalues of the 1^3 connected graphs on 1 to 6 vertices. Each graph is identified by P, Q, and the Harary number H. It is sufficient to tabulate the connected graphs, since the eigenvalues of any graph are the eigenvalues of its connected components. The eigenvalues in this table wer;e calculated on an IBM-360 computer, using MSDU, an IBM Scientific Subroutine. This subroutine approximates the eigenvalues of a real symmetric matrix, by the diagonalization method of Jacobi, modified by von Neumann. 146 TABLE A*2 EIGENVALUES OF THE 143 CONNECTED GRAPHS ON 1 TO 6 V E R T IC E S p £ H EIGENVALUES r " ‘ ... i 0 0 0.000 * ■ ? l 0 1 .000 -1 .000 f 3 2 0 1.414 0.000 -1 .414 3 . 3 0 2.000 -1.000 -1 .000 - :4 " 2 1. 61 e 0 . 61 n -0.61R -1 .618 i4 3 3 1.732 0.000 o.ooo -1 .7 3 2 14- 4 1 2. 170 0 .211 -1 .000 -1 .4 8 1 ‘4 4 2 2 .000 0.000 0.0 00 —.2.» 000 4 5 0 "'2 .'56"? “ G.C.OO - 1 .0 00 -1 .5 6 2 4 6 0 3.000 -1 .0 0 0 -1 .000 -1 .0 0 0 • 5 4 3 2.000 0,000 0,0 00 0 . 0 0 0 -2 .0 0 0 5 4 4 1.732 1 . 0 0 0 0 .0 00 -1 .0 0 0 -1 .7 3 2 5 4 6 1 .84 8 0.765 C.OCO -0.765 -1.848 ! 5 5 2 2.303 0.618 0.0 00 -1.303 -1.618 5 5 2. 136 0. 662 O.OCO - 0 ,6 6 ?_ -2 .1 3 6 r r “ 5 4 2 .214 l . C O O -0 .5 3 9 -1 .0 0 0 -1 .6 7 5 ‘5 5 c 2.343 0.471 0 . 0 0 0 -1.000 -1.814 '5 5 6 2 .000 0.618 0.618 -1.618 —i • 618 : 5 6 1 .56A _ l . C O O - 1 ..0 0 0 - l . O G O -1.562 15 6 2 2.666 0.335 o.ooo -1.271 -1 .7 4 9 j 5 6 4 2.481 0.689 0 . 0 0 0 -1.170 -2.000 5 6 5 2.449 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 -2 .4 4 9 5 6 6 2.641 0.724 - 0 . 5 8 9 -I j._Q.QP -1 .7 7 6 I s 7 1 2". 9 - 3 T “ 0.618 - G .4 63 -1 .4 7 3 -1 .6 1 8 ! 5 7 2 3 .000 0 . 0 0 0 0 .0 00 -1.000 -2.000 5 7 3. 086 0.428 - 1 . 0 0 0 - 1 .GCO -1 .5 1 4 | 5 7 4 2 .856 0 . 32 2 0 . 0 0 0 -1 .0 0 0 -2 .1 7 7 8 1 3 .'3 2 3 0 . 3 5 8 -1 .0 0 0 - l . O O C -1 .6 8 1 5 8 2 3.236 0 . 0 0 0 0.0 00 -1 .2 3 6 -2 . 00 0 ! 5 9 0 3.646 0. 000 -1 .0 00 - l . O f 0 — 1 • 64 6 1 5 10 0 4. COO -1 .0 0 0 -1 .000 -1.000 -1.000 1 .802 1.247 0.445 -0 .4 4 5 -1 .2 4 7 ■1.802 1 .902 1.176 0.000 0.000 - 1 . 176 -1 • 902 2.074 C . 63 5 0 .0 CO _O.COC -0.835. ■2.074 1 .932 T . f»0'f> "0.51 8" -O'. 5 18" -1.000 * 1•9 32 2 . 000 1 .000 O.OCO O.OCO -l.COO *2 .000 2 .236 0. 000 0.000 0.000 0 . 000 ■2.236 6 6 7 "2T c o r r T7DTOT- T T T T O ' f T "=T7 p £ H EIGENVALUES | 6 7 5 2 .414 1.000 0.414 -0.414 — . 000 -2.414 6 7 6 2 . 43 8 1. 13« 0 .6 1? - 0.020 — .618 -1 .757 6 7 7 2.39 1 C ^773 0.615 0.000 — .61 P -2.164 6 V ” « " 2.705'" 1 .056 0 .0 00 —0 • 5 60 . 350 — 1.851 6 7 9 2.65 5 1 .211 C.OCO - 1.000 . 00 0 - 1.666 6 7 10 2.625 1 .230 0.140 - 1.000 — .320 -1.673 6 7 11 2.754 0 .77 3 0.306 -0 . 6 09 — .32 9 -1. 8 94 6 /-- 12 2T539 IVGP'2’“ 0.~2'61“ -0.541 — .20 6 “-2. 136 ' 6 7 13 2 • 504 1 .264 0.0 00 -0.577 — .COO -2.191 6 7 14 2. 524 0.79 2 o.ooo 0 .OOO .792 -2.5 24 6 7 16 2.56? 1 . 000 0.000 O . 000 — . . 56 2 -2.COO 6 7. 17 ' 2 .555“ 0.6/7 G .000 O . 000 7 677 ~— 7. .558 ; 6 7 16 2.814 0 . 52 9 0.000 0 .000 — .343 - 2 .0C0 i 6 7 19 2.709 1. cco 0 .1 94 - 1.000 — .000 -1.903 i 6 7 20 2. 599 0 . 766 0.4 67 -0.3 65 — .30 5 -2.142 6 7 *1'" ... 2Y709 1 • CuO 0". 1 54 -1 .CCO ; c o o - 1‘. 903 6 7 22 2.791 0.616 0.000 0.000 — .61 8 -1.791 6 7 23 2.41 4 1 . 732 -0.414 -1 .0 00 — . GOO -1.732 6 7 24 2.732 l .con O.OCO -0.732 — .000 •.- 2.000 6 8 1 2. 947 1 .159 0.000 - 1.000 -1.286 -1.821 6 e 2 3.044 0.618 0.329 -0.548 -1.618 -1.824 6 8 3 3 .102 0.344 0.0 00 0.0 00 -1.32 3 -2.123 6 F ” 4 3.1/ / 0.676 ■ o ;ooo ---1 .OCO" -1,000" — 1 •8 56 : 6 8 5 2. 732 1.414 0.000 -0.732 -1.414 -2.OCO 6 8 6 2.741 0.710 0 .618 -0.231 -1.618 - 2.220 6 e 7 2. 791 l .ocr 0.618 -1 .CCO -1.618 -1.791 6 6 8 S. .433 O'27 2 T" O'. 3'09” —0•65/ -T . 00 0 "“-2; 31 2 * 6 6 9 2. 79 6 O.F53 0.000 0.000 -1.195 -2.454 6 8 10 2 .94 4 0.66 5 0.0 00 0 .000 -1.268 -2 . 2 40 6 8 11 3.014 0*648 0.1 °7 -0.7 25 -1.478 -1.856 6 8 "12 2.853" IVC 5 5 07183' —0.661 -1.272 —2.15F 6 e 13 2.89 5 1.00C- 0.0 00 -0.6C3 -1 .OOO -2.292 6 8 14 2.842 1.507 -0.5 07 - 1.000 ^-1.000 -1.84? 6 p 15 2. P2P c . nc o O .OOO 0.000 0.000 -2.P2P 6 "" 8 16 2.314 1 • GOO 0i5?9“ - I •oco —l • m 3 - 2.000 6 8 17 3 . C97 1 .117 -0.509 - 1.000 - 1.000 -1.704 6 8 20 2.981 1 .042 0.0 00 -0 . 706 -1.537 -1.780 6 8 21 2 .903 G . 80 6 0.000 0.000 -1.709 - 2.000 ”6*— ~8 _ “22 4. 164 ' PV 6 IF 0.227' —T. 000 "'-1.391 “ -1.618 ~ 6 8 23 2.73 2 G.73 2 0.0 cc 0 .000 -0.732 -2 . 7 32 : A e 24 3.04 8 0.621 o.ooo -0.756 - 1.000 -2.113 .... : 6 "9 ’ i '3. 372" G . COO 0 . 0 00 ■ 0 .000“~ -1 .000 -2 • 3 72 i 6 9 2 3.223 1 .GCO 0.1 12 -1.000 -1.527 -i .eo9 : 6 9 3 3.323 Ci . 3 5 P o.o no 0.000 '-1 . 6B 1 - 2.000 6 9 4 3 .404 0.490 0.251 - 1.000 -1.283 1-1.862 6 9~ 6 "3.281' o. n 2 (>. 0 00 —O .512 , -1.541 - 2.000 6 9 6 3.262 1 .340 - 1.0 00 - 1.000 -l.COO -1.602 6 9 7 3.000 1 .000 0.000 0.000 - 2 .GOO -2 .000 A 9 8 3.115 n.746 0.618 -0.861 -1.618 - 2.000 6 9 " " 9 " '”'3 .087."" 1 .15 6 ' ”0 VITO -1•OCO —1 . 174 -2.179 6 Q 10 3 . 09 2 0 . 702 0 . 0 (>0 0.000 -1 .286 -2.509 1 6 9 11 3 .182 1 .247 — 0 . 4 *+5 -G.594 -1.588 - 1.P02 6 . 9 12 3.354 1 .000 -0 .476 - 1 .OCO - 1.000 -1.8 77 6 *-9 - 13 ““ 3.169— “ 0.728" -- 0.2 8 0 ‘ —0.466 ’ -1 . 50 6 -2.205 ‘ 1 £ 9 14 3 .2.9 5 0.73 5 o .oco -0.59P -1.293 -2.139 1 £ 9 16 3 . 23 6 1 .000 0 .0 00 -1 .OCO -1.236 - 2.000 9 17 2. COO C.OGC 0.000 - 0 . cco o.coo -3.000 L : 1J+8 TABLE A.2— Continued p £ H EIGENVALUES T 6 9 18 3.236 0.610 0.618 -1.236 -1.618 *1.618 6 9 19 3.141 v I).485 C .000 O.OCO -1.000 -2.6 26 9 20 3.18° 0.835 0.0 00 -0.627 - 1.000 -2.396 1 £ 9™ 21 3'*~3B4 0 . /4? u .0 uu =170 00' -17328" -1.798 ! 6 10 1 3 • 59? 0.618 0 .1 59 -l.OCO -1.618 -1.752 6 10 2 3.449 0. 61 8 0.618 -1.449 -1.618 • 1.618 6 10 3 " 3.514 "■ 0 . 669 0 . G 00 = 07528" -1747 ft" -27177 . 6 10 4 3.534 1.083 -0.407 -1 .000 -1.511 ■1 .699 6 10 5 3.626 0.51 5 0.000 - 1.000 -J .000 -2.141 6 10 6 3.710 0.441 o.orc -1 .000 -1.384 -1.767 6 irr 7 3 • 8 ii • ft 0.16 7 =07555" -17 576"" -27247 6 10 8 ? . 49 8 0.730 0.151 ■-1.COO -1. 180 -2. 191 6 10 9 3 .690 0.753 -0.578 -1.000 -1 .000 -1.R65 6 10 10 3 .468 C.91 3 0.0 00 - 0 . 7 9 9 -1 .582 ■ 2.000 6 10 " 11 3.562” T . COO — C .5 62 -1 VO 00 -1 .000 -2.000 6 10 12 3 • 3°2 0.32 5 0.0 00 O.OOO -1.000 -2.718 6 10 14 3.37 2 1.000 O.OCO -1 .OCO -1 .000 -2.372 6 10 15 3.461 0.249 0.0 00 0 . 0 00 -1.339 -2.471 6 11 1 3.89 5 0.397 0.000 -1.000 -1.292 -2.000 6 11 2 3.828 1 .CCO •l.OCO -l.OCC -1 .0 0 0 -1 .828 6 11 3 3.778 0.71 1 0.000 -1.000 -1.48 9 • 2.000 ~b~ i r -*r "47C5"1~ "0748 3* 170*00 =T7CC0-— -1 . 00 O ' -1.534 6 n 5 3.859 0.779 ■0.379 -1.000 -1.476 -1 .7 83 6 li 6 3.766 0.000 O . O O O 0.000 -1.283 -2.484 6 n 7 3.732 0.414 0.2 68 -1.000 -1.00 0 •2.414 r> ii "ET 3 .'TTZT Trrciir "OTO'OO ------= 0 7 4 8 3 = " 1 7 61 8" -2.231 6 li 3.820 C.459 O.OCO -l.COO -1 .0 0 0 •2.279 6 12 1 4.119 0.619 -0.432 -1.000 -1.618 1.687 ; 6-- 12 “ 4. 162 n;ooo 0.0 00 "=1 .000“ —=1 .00 0“ 2.162 6 12 3 4.201 0.545 -l.OfO -1.000 -1.000 1.747 ! 6 12 4 4.068 0.36? 0.000 -1.000 -1.245 -2.185 j 6 12 5 4 . 00 0 O.OCO 0.000 0.000 -2.00 0 2;_000. 6 13 1 4.42 R 0.376 - 1 . 0 00 -1 .000 -1.000 —m1.804 6 13 2 4.372 C.000 0.0 00 - 1.000 -1.372 2.000 "6 14 ' 0 4. (TO 2 0 . 000 -1 .0(10 -1 • OCiO —1.000 17702 6 15 0 5.000 - 1.000 -1 .0 00 - 1.000 -1.000 — 1.000 I APPENDIX B DIAGRAMS OF REGULAR EXCEPTIONAL GRAPHS A graph is exceptional if its least eigenvalue is greater than or equal to -2 , but it is not a generalized line graph. A graph is regular if all vertices have the same degree. Bussemaker, Cvetkovic, and Seidel (1976) found, in a computer-aided search, that there are exactly 187 regular exceptional graphs. Their results are summarized in Table B.l. A regular exceptional graph on p vertices has one of three possible degrees* either &p-2, or (2/3)p-2, or (3A ) p - 2. The number of each type is shown in the table. Asterisks mark the graphs discovered earlier, by Petersen, Hoffman and Ray-Chaudhuri, Shrikhande, Clebsch, Schl&fli, and Chang. Some regular exceptional graphs are shown in Fig. B.l. The last six graphs are indicated by their'"Seidel switch diagrams." First, all vertices along a double line are adjacent to each other. Second, the vertices are in two subsets (• and 0)1 two vertices in the same subset have 1^9 150 TABUS B.l THE 187 REGULAR CONNECTED EXCEPTIONAL GRAPHS 1 [Number of Degree Vertices P ip-2- (2/3)p-2 <3A)p-2 ♦Remarks (v,k,^) 8 1 * T(C^). 9 2 . . • - ■ • . 10 5 * Petersen graph. O C * * i 12 5 1 T(C^), non-(3,2,2). j i Also, the Hoffman & j Ray-Chaudhuri graph, non-( 4 * , 3,2). 21 1 6 Shrikhande graph, j !1 16 35 * non-L(K^). | 1 * Clebsch. graph. i ■ 18 38 * * non-L(K^ ^ ) . • 20 27 i 21 2 22 18 2k 8 1 26 0 ] i 27 i * Schl&fli graph. 28 * 3 Chang graphs, non-L(Kg;• Total 163 21 3 Grand total* 187 SOURCE* Bu'ssemaker,. Cvetkovid, "and Seidel (1976). Petersen graph i A / \ * .« .J The five cubic (degree 3) regular exceptional graphs — 0 I \ /K i \ I '■ / Total graph Total graph T(C^)t Hoffman and Ray-Chaudhuri non-BIBD(3» 2,2) (1965 a), non-SBIBD (4,3,2 :© r--f. ? 0 —— L*. Qrr~. 1 . O - —© 1 li - J r - t =o_^® I- - O I 4 - 4 ^ Shrikhande (1959 b), Clebsch (1868) non-L(Kg^) non-L(K/f^) r © ~_ ’o nI. V I - '! 1 O- -©™o= I: r ■A H © ©- [l \\ I? 0 |! © i1 11 F h TT Ti j * r ©^- ©.-=-© ..©-.i..#. ^0 u—I |r~ ------©--© ©- -O-— o The three Chang (i960) graphs, non-L(Kg) Fig. B.l. Some regular exceptional graphs 152 adjacency as drawn, but two vertices in different subsets are adjacent in the graph if they are nonad.iacent in the diagram. Four regular exceptional graphs are represented in the root system Eg (Bussemaker, Cvetkovic, and Seidel 1976). They are* the Petersen graph, and the three regular exceptional graphs of degree (3/*0 p-2 , with p - 8 , 12, or 16 vertices. Figures B.2 - B.5 show the graphs, together with their base graph diagrams, as defined in Chapter V. The abbreviation "BCS # 5'* denotes graph number 5» in Table 9*1 of Bussemaker, Cvetkovic, and Seidel (1976, pp. 58-71)* In each base graph B(Hiab), a =» 1 and b - 2 . Base graph diagram Fig. B.2. BCS # 5* “the Petersen graph Base graph diagram Fig. B.3 . BCS # 185» the total graph T(C^) 153 Graph 12 Base graph diagram ■ /1 (x7 Y9)-^C8 Yio)— . ; 11 i -6,^> T \f7■ ' ' z 5 r -Z3; --J •2-— -8 ■ Z6 Fig. B.^. BCS # 186 Seidel switch diagram' Base graph diagram JL.... 11 @ iX^ .. 7 2 1^ (&) ' ' / (Z5 ;----Z^> 23 )---- T15/y 3 15 @ ‘v§) 6 16 C^y v 5 f 28 ; Z7 ) Z6 ,---- T161 y Fig. B.5* BCS # 18?t the Clehsch graph Twenty-four regular exceptional graphs are represented in the root system E^ and not in (Bussemaker, Cvetkovic, and Seidel 1976). They are graphs BCS # 1, 9, 69, and all twenty-one of the graphs BCS # l6b-l8k of degree (2/3)p-2, with p = 9, 12, 15* 18, 21, or 2? vertices. Figures B .6 - B.9 show four of these graphs, and the corresponding “base graph diagrams, as defined in Chapter V. 154 Graph Base graph diagram Pig. B.6. BCS # 1 Graph Base graph diagram -> T 1 1 \Z3. T7— T9 y - 7 ! \ 1C 0 / 1 1 r 1 // Z4 — )-- :(X6 Y 5 )------T 12 Pig. B.7. BCS # 9, the Hoffman and Ray-Chaudhuri graph 155 Seidel switch diagram Base graph diagram T16 G > ~ 9 11 13 p ) < N 6 J) 12 14 7“ 10 © 15 1 2 3 ,1& / A ■ Z4 — X8 ;(X10 Y7).. - Fig. B.8. BCS # 69» the Shrikhande graph Segre diagram Base graph diagram 2 T A 2 3 26 ! y ; 25 6 17 22 24 5 I ' ' 4 9 16 21 23 8 1 A . . / ? / 18 13 10 19 14 11 .L. ' \\ \ ... 20 15 12 Z9 r I Z6 A X18 Y13/--- 7 T27\/» A Z12 Z8 - Z5 1X19 X14----- \-T26 // / i-.N \ w Z10; Zll Z7 . z4 - ' X20 Y15;------T25 Fig. B.9. BCS # 184, the Schiafli graph APPENDIX C DIAGRAMS OF NON-REGULAR EXCEPTIONAL GRAPHS The root system Eg has hase graph L(K^ ^), as defined in Sections 3 and 4 of Chapter V. The ^2 patterns of X, Y, and W in Eg are shown in Table Jf.l of Chapter V. In Figures C.l - C.12, we have drawn some base graph diagrams, and constructed the corresponding graphs. By Lemmas 3.2 - 3*^ and Theorems ^.3 - of Chapter III, none of these graphs is a generalized line graph. Thus they are exceptional graphs. Each exceptional graph that has previously appeared in the literature is regular (see Appendix B). Therefore, all of the graphs in Figures C.l to C.12 are new exceptional graphs. In each base graph B(H:ab), a - 1 and b = 2 . The symbol W is expanded to* CX y ) . Each vertex T is joined by a solid line to one row or column of the base graph, and is joined by a dashed line to another row or column of the base graph. A vertex T is adjacent to a vertex X or Y if they are joined by a solid line, A vertex T is adjacent to a vertex Z if they are joined by a dashed line. A vertex T is nonadjacent to a vertex if they are joined by no lines or by two lines. 156 157 Graph Base graph diagram 1 3 (Y3/'fxiJ- ) 1 V V 2 ^ , I , X^./- — T 6 5 6 . P-. / J— * i_/ Fig. C.l. Exceptional graph — —•/ Graph Base graph diagram 3 (Y3)-' i- x y / Fig. C.2. Exceptional graph Graph Base graph diagram 1\ I J>3 2.r / T 6 6 ---- 5 I Fig. C.3» Exceptional graph Graph Base graph diagram ;5 6 ' 1 -1 Z 5/ - n r" r t v • - ■” /” Jt6 Fig. C.^. Exceptional graph Graph Base graph diagram 1 i ^X3y- x* y ~ 2 .':r' r 3-4 5—6 (. r x /-.X5> X Fig. C.5. Exceptional graph Graph Base graph diagram 1 X6 4— 5— 6 Fig. C.6. Exceptional graph Graph Base graph diagram 1 K/K.I (X3v yX4 yX5 „ X .-6 X6 ; r ':r' V A - Fig. C.?. Exceptional graph Graph Base graph diagram X6 H ,xZ A _ / \ Pig. C.8 . Exceptional graph Graph Base graph diagram X3 Kx'tK X5; r V ^•5 >X6 H x? H j 'rr" 'tr' - r (_ \ Fig. C.9 . Exceptional graph Graph Base graph diagram 3 x I i ( x k Y 3V x5 X6 h 2, ■x r v 5 * 0 : 6 X7 T- "I' Fig. C.10. Exceptional graph 160 Base graph diagram ( x4~Y3>- . X5j~ \ X6 j T ^ w (^ \ r 5“ 7— 8 v s ^ 0 " 0 Fig. C.ll. Exceptional graph Graph Base graph diagram X3 -i'xiT H x T ) L J fx6 >-(X7 V ( X 8 ) A T J~. ^--(xioh-'xii) 9;— 1 Fig. C.12. Exceptional graph BIBLIOGRAPHY 1. M. So Aigner, A characterization problem in graph theory, J. Combinatorial Theory 6 (1969 a )» **5-55- 2. M. S. Aigner, The uniqueness of* the cubic lattice graph, J. Combinatorial Theory 6 (1969 b), 282-297. 3. M. S. Aigner and T. A. 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