Graphs with Least Eigenvalue −2: Ten Years On

GRAPHS WITH LEAST EIGENVALUE −2: TEN YEARS ON

DragoˇsCvetkovi´c MatematiˇckiInstitut SANU Kneza Mihaila 36 11001 Beograd, p.p. 367 Serbia Email: [email protected]

Peter Rowlinson1 Mathematics and Statistics Group Institute of Computing Science and Mathematics University of Stirling Scotland FK9 4LA Email: [email protected]

Slobodan Simi´c State University of Novi Pazar Vuka Karadˇzi´cabb 36 300 Novi Pazar Serbia and MatematiˇckiInstitut SANU Kneza Mihaila 36 11001 Beograd, p.p. 367 Serbia Email: [email protected]

Abstract The authors’ monograph Spectral Generalizations of Line Graphs was published in 2004, following the successful use of star complements to complete the classiﬁcation of graphs with least eigenvalue −2. Guided by citations of the book, we survey progress in this area over the past decade. Some new observations are included.

AMS Classiﬁcation: 05C50

Keywords: graph spectra, Hoﬀman graph, signed graph, signless Laplacian, star complement.

1Correesponding author: tel.:+44 1786 467468; fax +44 1786 464551 1 Introduction

Let G be a simple graph with n vertices. The characteristic polynomial det(xI − A) of the adjacency matrix A of G is called the characteristic polynomial of G and denoted by PG(x). The eigenvalues of A (i.e. the zeros of det(xI − A)) and the spectrum of A (which consists of the n eigenvalues) are also called the eigenvalues and the spectrum of G, respectively. The eigenvalues of G are real because A is symmetric; they are usually denoted by λ1, λ2, . . . , λn, in non-increasing order. The largest eigenvalue λ1 is called the index of G, and G is said to be integral if every eigenvalue is an integer. An overview of results on graph spectra is given in [2], [5] and [1]. Let L (L+, L0) be the set of graphs whose least eigenvalue is greater than or equal to −2 (greater than −2, equal to −2). A graph is called an L–graph (L+–graph, L0–graph) if its least eigenvalue is greater than or equal to −2 (greater than −2, equal to −2). The line graph L(H) of any graph H is defined as follows. The vertices of L(H) are the edges of H, and two vertices of L(H) are adjacent whenever the corresponding edges of H have a vertex of H in common. Interest in the study of graphs with least eigenvalue −2 began with the elementary observation that line graphs have least eigenvalue greater than or equal to −2. A natural problem arose to classify the graphs with such a remarkable property. It transpired that line graphs share this property with generalized line graphs and with some exceptional graphs (defined below). The authors’ scientific monograph on L–graphs [4] was published in 2004, following the successful use of star complements to complete the classification of L-graphs, and it summarized almost all results on the subject known at the time. We felt that the main problems concerning L–graphs had been solved. However, we now see that the book did not mark the end of the story: it is the aim of this paper to review the many new and important results on star complements and spectral aspects of L–graphs that have been obtained in the last decade. The rest of the paper is organized as follows. Section 2 contains some technical details concerning the book and related bibliographies. In Section 3 we present some definitions and basic results required subsequently. Sections 4 and 5 contain recent results obtained by the star complement technique. In Section 6 we describe constructions for the exceptional regular L-graphs, 1-Salem graphs and non- bipartite integral graphs with index 3. Section 7 deals with spectral characterizations and cospectral graphs. In Section 8 we discuss Hoffman graphs and limit points for the least eigenvalue of a graph in the interval [−3, −1]. Section 9 is concerned with the multiplicity of 0 as an eigenvalue of an L-graph. In Section 10 we point out the relation between the signless Laplacian spectrum of a graph G and the (adjacency) spectrum of L(G). Section 11 shows how results on L-graphs have been extended to signed graphs. Section 12 deals with applications to control theory and computer science, while Section 13 summarizes developments in a miscellany of other topics.

2 Corrections, reviews and citations

The Appendix contains a list of errata in [4], taken from the website www.cs.stir.ac.uk/~pr/, where a list of corrections is currently maintained; some of these corrections were already given in [41]. The ﬁrst section in our bibliography below gives a list of complete references that were not given fully in [4]. References [BeSi] and [CvSt] from [4] refer to [4] itself, while the paper [144] cites [4] as a manuscript. The following reviews of the book have been published: P. J. Cameron, Bulletin of the London Mathematical Society 37 (2005), 479-480; M. E. Watkins, Zentralblatt 1061.05057; M. Doob, Mathematical Reviews, 2005m: 05003. According to Google Scholar the book has been cited over 100 times. Papers that cite the book are included in the section of our bibliography entitled ‘New papers’. This section includes some papers that do not cite [4] but contain new results on star complements, spectral aspects of L–graphs and various generalizations. We also give details of three relevant doctoral theses [10, 11, 12] which have been defended in the period under review. The last section of the bibliography contains other

1 references (mainly published before the book) which are necessary for explanations. Our book [4] has been cited as a general reference in the monographs [8, 9], and in the papers [13, 20, 22, 28, 39, 47, 50, 54, 57, 59, 68, 69, 70, 71, 81, 113, 115, 125, 126, 127, 128].

3 Preliminaries

In this section we present some relevant definitions and basic results. We write E(λ) for the eigenspace of the eigenvalue λ. The maximum and minimum degrees in a graph G are denoted by ∆(G) and δ(G) respectively; the vertex-connectivity of G is denoted by κ(G). Recall that the cone over a graph G is the graph obtained from G by adding a new vertex adjacent to all vertices of G.A(κ, τ)-regular set in a graph G is a vertex subset S which induces a κ-regular subgraph such that every vertex not in S has τ neighbours in S. (Thus if G is a regular graph, then a (κ, τ)-regular set S determines an equitable bipartition in G.) The cocktail-party graph mK2 is denoted by CP (m). A generalized cocktail-party graph is obtained from a complete graph by deleting some independent edges. A generalized line graph L(G; a1, . . . , an) was originally defined (in [149]) for graphs G with vertex set {1, . . . , n}, and non-negative integers a1, . . . , an, by taking the graphs L(G) and CP (ai)(i = 1, . . . , n) and adding extra edges: a vertex e in L(G) is joined to all vertices in CP (ai) if i is an end- vertex of e as an edge of G. We include as special cases an ordinary line graph (a1 = a2 = ··· = an = 0) and the cocktail-party graph CP (m)(n = 1 and a1 = m). We use the abbreviation GLG for a generalized line graph. As noted in [141], it is convenient to regard a GLG as the line graph of a B-graph (‘graph with blossoms’), which is a multigraph defined as follows. First, a pendant double edge is called a petal, while a blossom Bm consists of m petals (m ≥ 0) attached to a single vertex. (An empty blossom B0 has no petals and is reduced to the trivial graph K1.) Now a B-graph is a multigraph H obtained from a graph G by adding a (possibly empty) blossom at each vertex. If ai petals are added at vertex i (i = 1, . . . , n) then we write H = G(a1, . . . , an) and L(H) = L(G; a1, . . . , an). We require that in L(H) two vertices are adjacent if and only if the corresponding edges in H have exactly one vertex in common. We refer to H as a root graph of L(H). Note that L(Bm) = CP (m). A proper generalized line graph is a GLG which is not a line graph. An exceptional graph is a connected L-graph which is not a GLG. A generalized exceptional graph is an L-graph with at least one exceptional component. The set of regular exceptional graphs is partitioned into three layers according to degree r and order n as follows [4, Theorem 4.1.5]:

1. n = 2(r + 2) ≤ 28,

3 2. n = 2 (r + 2) ≤ 27 and G is an induced subgraph of the Schläfligraph, 4 3. n = 3 (r + 2) ≤ 16 and G is an induced subgraph of the Clebsch graph. There are 163 graphs in the 1st layer, 21 in the 2nd layer and 3 in the 3rd layer, i.e. 187 in total. We say that an L-graph, with adjacency matrix A, is representable in a set S ⊆ IRt if A + 2I is the Gram matrix of some vectors from S; note that t may be taken as the codimension of E(−2). As explained in [4, Section 3.6], an exceptional graph is representable in the root system E8. By a spectral graph theory we understand, in an informal sense, a theory in which graphs are studied by means of the eigenvalues of some graph matrix M. This theory is called M-theory. Hence there are several spectral graph theories (for example A-theory, based on the adjacency matrix). Of course, spectral graph theory consists of all these special theories, including their interactions. Let G be a finite set of graphs, and let G0 be the set of graphs in G which have a cospectral mate in G with respect to M. The ratio |G0|/|G| is called the spectral uncertainty of (graphs from) G with respect to M.

2 4 Star complements

Let G be a graph of order n with µ as an eigenvalue of multiplicity k, and let t = n − k. Thus the eigenspace E(µ) of a (0, 1)-adjacency matrix A of G has dimension k and codimension t.A star set for µ in G is a subset X of the vertex-set V (G) such that |X| = k and the induced subgraph G − X does not have µ as an eigenvalue. In this situation, G − X is called a star complement for µ in G. It is shown in [3, Theorem 7.1.3] that for any graph G, with distinct eigenvalues µ1, . . . µm, there exists a partition V (G) = X1 ∪˙ · · · ∪˙ Xm such that Xi is a star set for µi (i = 1, . . . , m); such a partition is called a star partiton for G. The basic properties of star sets and star complements are established in [4, Chapter 5]. The fundamental result is the following, where for any X ⊆ V (G), we write GX for the subgraph of G induced by X. We take V (G) = {1, . . . , n}, and write u ∼ v to mean that vertices u and v are adjacent. Theorem 4.1. (See [4, Theorem 5.1.7].) Let X be a set of k vertices in G and suppose that G has A B> adjacency matrix X , where A is the adjacency matrix of G . BC X X (i) Then X is a star set for µ in G if and only if µ is not an eigenvalue of C and

> −1 µI − AX = B (µI − C) B. (1)

x (ii) If X is a star set for µ then E(µ) consists of the vectors (x ∈ IRk). (µI − C)−1Bx

Let H = G − X, where X is a star set for µ. The columns bu (u ∈ X) of B are the characteristic vectors of the subsets ∆H (u) = {v ∈ V (H): u ∼ v} (u ∈ X). For u, v ∈ X, we have from Eq.(1):

 µ if u = v > −1  bu (µI − C) bv = −1 if u ∼ v (2)  0 if u 6∼ v

Eq.(1) shows that G can be constructed from the eigenvalue µ, the star complement H and the H-neighbourhoods ∆H (u)(u ∈ X). To find the graphs with prescribed µ and H we consider the compatibility graph Γ(µ; H), defined as follows in view of Eq.(2). The vertices are the (0, 1)-vectors b ∈ IRt such that b>(µI − C)−1b = µ, with b ∼ b0 if and only if b>(µI − C)−1b0 ∈ {−1, 0}. Thus the required graphs correspond to cliques in Γ(µ; H). An exceptional graph has an exceptional star complement for −2 [4, Theorem 5.3.1], and there are just 573 possible star complements: 20 have order 6, 110 have order 7 and 443 have order 8 (see [4, Table A2]). Thus every exceptional graph is determined by a clique in some Γ(−2 : H), where H is one of 573 possible graphs H. In fact, the maximal exceptional graphs can be found as extensions of a star complement of order 8 [4, Proposition 5.3.2]; these star complements are labelled H001,...,H443 in [4]. Accordingly the maximal exceptional graphs were found by using a computer (i) to search for the maximal cliques in 443 graphs Γ(−2 : H), (ii) to check for isomorphisms. The authors remain indebted to M. Lepović for undertaking the computing. There are exactly 473 maximal exceptional graphs, and the number of their vertices varies between 22 and 36. They are described in Chapter 6 and Table A6 of [4] where they are denoted by G001,...,G473. Most (432 of them) are 29-vertex cones of the form G(P ), where G(P ) denotes the cone over the graph obtained from L(K8) by switching with respect to the edge set of a spanning subgraph P of K8. Theorem 4.2. Let G be an exceptional graph. Then there exists a graph G0 such that G is an induced subgraph of G0 where either (a) G0 has H440 as a star complement for the least eigenvalue −2, or (b) 0 G is the cone over L(K8). The graph H440 is the complement of the unique forest with vertex degrees 3,2,1,1,1,0,0,0. The theorem follows from the computer search but there is always a tendency in mathematics to prove theorems without recourse to a computer. A sketch of such a proof is already given in reference

3 [Cve13] of [4], while some additional details appear in [30] and [37]. Consider a maximal exceptional graph G of the form G(P ) for some 8 vertex graph P . We want to prove that G(P ) contains H440. To that end let x be a vertex of degree 7 in H440; the vertex x will coincide with the ‘top’ of the cone G(P ). Using brute force we can check that the switching class of H440 − x contains the line graphs of 11 particular graphs on 8 vertices. In each case we find the corresponding switching sets. Now all possible graphs P should be considered, but this classification was not completed in [30]. In the remainder of this section we describe how star complements have since been used in wider contexts. Let G be a graph with H as a star complement for an eigenvalue µ 6∈ {−1, 0}. It is shown in [93] that for each k ∈ IN there exists a smallest non-negative integer f(k) such that κ(G) ≥ k whenever κ(H) ≥ k and δ(G) ≥ f(k); moreover, f(1) = 0, f(2) = 2, f(3) = 3, f(4) = 5, f(5) = 7 and f(6) ≥ 8. To see that f(5) ≥ 7 and f(6) ≥ 8 one can exploit [4, Example 5.1.14] to construct suitable graphs G1,G2 with K8 as a star complement H for −2 [93, Examples 4.2, 4.3]. For G1 we add to H four pairwise adjacent vertices whose H-neighbourhoods are the four 3-subsets of a 4-set in V (H); then κ(H) = 7 and δ(G1) = 6, while κ(G1) = 4. For G2 we first add a set X of ten vertices whose H-neighbourhoods are the ten 3-subsets of a 5-set in V (H); secondly we add an edge between vertices u, v of X whenever |∆H (u) ∩ ∆H (v)| = 2. Now X induces L(K5), κ(H) = 7 and δ(G2) = 7, while κ(G2) = 5. We have seen that a graph G is determined by an eigenvalue µ, a corresponding star complement H = G − X and the H-neighbourhoods ∆H (u)(u ∈ X). This property is the key to a number of characterizations of graphs by star complements [87, 91, 97, 98, 114, 130]. Note that for an r- regular graph, µ is non-main when µ 6= r and so Theorem 2.1(ii) provides the additional condition > −1 t bu (µI − C) j = −1 (where j is the all-1 vector in IR ). We give two examples of characterizations of regular graphs. First, the Hoffman-Singleton graph is the unique regular graph having the tree shown in Fig. 1 as a star complement for the eigenvalue 2 [97, Theorem 2.3]. Secondly, a strongly regular graph with parameters (81, 20, 1, 6) is easily seen to have a windmill of order 21 as a star complement for 2, and this is sufficient to establish the uniqueness of such a graph [114, Theorem 7]. The regular graphs with a windmill as a star complement for some positive eigenvalue µ are investigated in [98]: it is shown that such graphs are necessarily strongly regular with parameters ((µ2 +3µ−1)2, µ2(µ + 3), 1, µ(µ + 1)). In the reverse direction, Biggs [21, Theorem 6.1] shows that if G is a strongly regular graph with parameters (n, r, e, f) such that f = µ(µ + 1) and µ is the positive eigenvalue 6= r, then the closed neighbourhood of a vertex is a star complement for µ.

QPXPXX QPPXXX t Q PP XXX ¢B ¢B ¢B ¢B ¢B ¢B ¢B ¢ tB ¢ tB ¢ tB ¢ tB ¢ tB ¢ tB ¢ tB tttttttttttttt Fig. 1: A star complement for the eigenvalue 2.

On the other hand, star complements can be used to show that there is no strongly regular graph with parameters (57, 14, 1, 4) [76, Theorem 4]. This last result requires (i) inspection of several possibilities for a star complement H for the eigenvalue 2, (ii) a computer search for maximal cliques in the associated compatibility graphs. In [90] it is shown that, aside from the complete multipartite graphs and graphs of Steiner type, there are only finitely many connected strongly regular graphs with a regular star complement of prescribed degree. If G is an extremal strongly regular graph with µ as the eigenvalue of largest multiplicity then t = 4µ2 + 4µ − 2 (see [87]), and by interlacing G has independence number at most t − 1. If G has an independent set S of size t − 1 then for any vertex v 6∈ S, the subgraph induced by S ∪˙ {v} has 2 the form K1,s ∪˙ (t − s − 1)K1. As long as µ 6= s, this graph is a star complement for µ. These observations are used in [87] to show that the Schläfligraph and the McLaughlin graph are the only graphs that arise (with µ = 1, µ = 2 respectively).

4 It follows from Eq.(1) that if X an independent star set for a non-zero eigenvalue µ then |X| ≤ t. If |X| = t then (by interlacing) µ is the least eigenvalue of G unless µ = 1 and G = tK2. It is shown in [94, Theorem 3.2] that if X is an independent star set for −2 of size t then either (a) t = 7 and G is the incidence graph of the complement of the Fano plane, or (b) t = 8 and G is one of thirteen graphs (of order 16). The graphs in which every star set is independent (or in which every star set induces a clique) are investigated in [15]. In particular, among connected graphs of order > 3 with just three distinct eigenvalues, only stars have a star partition in which each cell is an independent star set [15, Theorem 5], and only the graphs Kr,r (r ≥ 2) are such that every star set induces a clique [15, Theorem 7]. If G is a graph with second largest eigenvalue λ2(G) ≤ 1 then either (a) the least eigenvalue of G is not less than −2, or (b) G has exactly one eigenvalue less than −2 [4, Theorem 7.1.1]. Case (b) is not yet resolved, but there are some partial results embracing both cases. In particular, if H is a star complement for 1, the possibilities for H are identified when H is unicyclic [107, 112], a tree [106] or a complete graph [106]. Moreover, there are just 14 possibilities for H when H has order at most 5 [107, Theorem 5]. Further results may be found in [62, 108, 110]. Soon after star complements had been exploited to find the maximal exceptional graphs, they were used to establish a sharp upper bound for the multiplicity k of a graph eigenvalue µ 6∈ {−1, 0} [135]: t t if t > 2 then k ≤ 2 ; moreover if µ is non-main then k ≤ 2 − 1. The second bound is attained in every extremal strongly regular graph. The first bound is attained in 3K2 (with µ = 1), and in the unique exceptional graph of order 36 (labelled G473 in [4]). Note that it follows immediately from the first bound that an exceptional graph has order at most 36 because in this case an exceptional star complement for −2 has order t ≤ 8. (This result was proved previously by other means in [138].) t It remains an open question whether G473 is the only connected graph in which the bound of 2 is attained, and this is the subject of the next section. Here we note that star complements have been used to establish sharp upper bounds for eigenvalue multiplicity in further classes of graphs, including trees [88], cubic graphs [95], and graphs with prescribed girth [89]. Some results from [88] have been refined in [46].

5 The exceptional graph of order 36

The graph G473 is obtained from L(K9) by switching with respect to an 8-clique; its spectrum is (7) (28) ∼ 21, 5 , −2 . We may take an 8-clique as a star complement for −2, say H = G − X = K8, and then the H-neighbourhoods of vertices in X are the twenty-eight 6-subsets of V (H). The bipartition X ∪˙ X (where we write X for V (H)) is equitable. The vertices in X have degree 18, those in X have degree 28, and the graph induced by X is L(K8). After 12 years it remains an open problem to determine all the connected graphs with an eigenvalue t µ 6∈ {−1, 0} of comultiplicity t and multiplicity 2 . In the absence of a solution one can ask restricted questions such as: (1) Is G473 the only such graph that can be obtained from a strongly regular graph by switching with respect to a star set X having the property that X ∪˙ X is equitable? (2) Is G473 the only such graph with a complete star complement? The ﬁrst question is answered aﬃrmatively in [92]. The second question remains open but we provide a partial answer below. Note that since a complete star complement is devoid of structure, the existence of an appropriate star set X depends solely on combinatorial considerations. ∼ We use the notation of Theorem 4.1, with H = G − X = Kt (t > 2) and µ 6= 0. Note that µ 6= −1 or t − 1, and C = J − I, where J is the all-1 matrix of size t × t. Thus 1 1 (µI − C)−1 = I + J. µ + 1 (µ + 1)(µ + 1 − t)

If u ∈ X and |∆H (u)| = s then, from Eq.(2) (with u = v) we obtain

s2 − (t − µ − 1)s + µ(µ + 1)(t − µ − 1) = 0. (3)

5 1 p 2 It follows that s = 2 (q ± q − 4µ(µ + 1)q), where q = t − µ − 1 6= 0. Thus in general there are two possibilities for the size of an H-neighbourhood. If all H-neighbourhoods have the same size then we can obtain the desired outcome: Theorem 5.1 Let G be a ﬁnite graph of order n > 3, with X as a star set for the eigenvalue ∼ 1 µ 6∈ {−1, 0}. Suppose that H = G − X = Kt and |∆H (u)| = s for all u ∈ X. If n = 2 t(t + 1) then µ = −2, t = 8, s = 6 and G is the graph G473.

Proof. If u, v are distinct vertices of X such that |∆H (u)| = |∆H (v)| = s then from Eq.(2) we have:

( s2 q if u is not adjacent to v, |∆H (u) ∩ ∆H (v)| = s2 (4) q − µ − 1 if u is adjacent to v.

1 Note that G is connected because H is connected and s 6= 0. We assume that n = 2 t(t + 1) > 3. If n = 6 then, by inspecting the spectra of connected graphs in [142], we see that that no example t arises. Thus t ≥ 4 and |X| = 2 > t; we deduce that µ is an integer. It follows from Eq.(4) that if t 4 ≤ s ≤ t − 2 then the 2 neighbourhoods ∆H (u)(u ∈ X) form a tight 4-design D [139, Theorem 1.52]. Such designs are extremely rare; indeed, if 4 ≤ s ≤ t − 3 then D or its complement is the Steiner system S(4, 7, 23) [139, Theorem 1.54]. These two possibilities are excluded by Eq.(3): if (n, s) = (23, 7) or (23, 16) then there is no integer µ for which Eq.(3) is satisﬁed. Accordingly, it remains to consider the cases s = 2, s = 3 and s = t − 2. We make two preliminary remarks. First, µ < t because the graph obtained from H by adding one vertex from X is a proper subgraph of Kt+1; hence q > 0. Secondly, X contains both adjacent and non-adjacent pairs of vertices, for otherwise |∆H (u) ∩ ∆H (v)| takes the same value on all pairs u, v ∈ X, and then we have |X| ≤ t by [139, Theorem 1.51], a contradiction. If s = 2 then Eq.(3) yields q(2 − µ2 − µ) = 4, whence µ2 + µ < 2, a contradiction. Next suppose that s = 3. Then q(3 − µ2 − µ) = 9, whence µ ∈ {−2, 1} and q = 9. If µ = −2 then t = 8, and from Eq.(4) we see that the neighbourhoods ∆H (u)(u ∈ X) form an intersecting family of 3-sets. Now 7 by Theorem 1 of [136, Chapter 7], we have |X| ≤ 2 , a contradiction. If µ = 1 then we obtain a contradiction from Eq.(4) in respect of any pair of adjacent vertices in X. Finally suppose that s = t − 2. Then Eq.(3) becomes

(µ2 − 1)(s − µ) = −2µs.

We deduce that µ divides s. Also, 2µs 6= 0 and µ2 − 1, µ are coprime. Hence µ2 − 1 divides 2s and we may write 2s = µ(µ2 − 1)b. We obtain (µ2 + 2µ − 1)b = 2, whence µ ∈ {−3, −2}. If µ = −3 then Eq.(3) has no positive solution for t. On the other hand, if µ = −2 then t = 8 and the neighbourhoods ∆H (u)(u ∈ X) are precisely the 28 subsets of V (H) of size 6. Thus the only graph that arises is the graph G473. This completes the proof of Theorem 5.1.

In Theorem 5.1, the hypothesis that all H-neighbourhoods have the same size is equivalent to the requirement that V (H) is a regular set (explicitly, a (t − 1, s)-regular set). In [16] the authors investigate the general situation of a star complement H for which V (H) is a regular set, and they provide a number of examples. Question (2) above concerns the graphs of largest possible order with a complete star complement. The analogous question for bipartite graphs is addressed in [96], and we mention two results from this paper. Let G be a bipartite graph with µ as an eigenvalue of multiplicity k > 1. If G has the complete bipartite graph Kr,s (1 ≤ r ≤ s) as a star complement for µ then k ≤ s − 1, with strict inequality for large enough s when µ is non-main. When r = 1, k = s − 1 and µ is non-main, G can be constructed from the incidence graph of a certain symmetric 2-design D = D(µ) by adding adjacent vertices v, w such that v is adjacent to all points of D and w is adjacent to all blocks of D. Five examples with spectrum ±3(56), ±15 arise when µ = 3, s = 56 and D is taken to be a biplane of order 9, i.e. a symmetric 2-(56,11,2) design.

6 6 Constructions

In this section we discuss recent constructions for three interesting sets of L-graphs: regular exceptional graphs, non-bipartite 1-Salem graphs and non-bipartite integral graphs with index 3. We first describe a recent recursive construction [19] of the regular exceptional graphs, based on the new (κ, τ)-extension technique suggested in [24]. The graphs in question are completely described in [4, Table A3]. Let G be a regular graph of degree r and order n. Let α(G) be the size of a largest independent set in G and let λn be the least eigenvalue of G. Then −nλ α(G) ≤ n . r − λn This inequality is known as the Hoffman inequality although A. J. Hoffman never published it. For some bibliographical details related to this bound, see [24]. The bound is attained if and only if G has a (0, τ)-regular set with τ = −λn (see [26, 148]). In [24] it is noted that in the case of regular exceptional graphs in the 1st and 2nd layers, the Hoffman bound is attained with α(G) = 4, α(G) = 3 respectively. (This is an empirical observation made once the regular exceptional graphs are known.) Thus if G is a graph in the first or second layer then G has a (0, 2)-regular set S, of size 4 or 3 respectively. In this situation, G − S is an (r − 2)-regular L-graph, and we say that G is a (0, 2)-extension of G − S. It follows that G can be constructed from successive (0, 2)-extensions of a regular line graph L. In this way we obtain an algorithm for the construction of graphs in the first and second layers. In practice, it suffices to take L ∈ {3K2, 2C4,C3 ∪˙ C5,C8} to obtain all graphs in the first layer, and L ∈ {C6, 2K3} to obtain all graphs in the second layer. Let us describe (0, 2)-extensions of size 4 in some detail. Consider an extension G0 of a regular graph G by an independent set S = {1, 2, 3, 4}. Each vertex of G should become adjacent to exactly two vertices of S. We now define an r-regular multigraph M(S) with S as its vertex set. If a vertex v of G becomes adjacent to vertices x, y of S, then there is an edge labelled v between x and y in M(S). In this way, the vertices of G determine a partition of the edges of M(S). There are six 2-element subsets of S, and the construction of G0 requires a partition of V (G) into six subsets which are assigned to 2-element subsets of S in such a way that M(S) is regular of degree r. However, G0 need not be an L-graph, and so this must be checked in the actual constructions. The multigraph M(S) can be associated with a weighted complete graph on the four vertices of S (see Fig. 2). The weight xi on the i-th edge of the multigraph M(S) represents the number of vertices of G adjacent to corresponding vertices in S.

2 ¡A ¡ tA ¡ A ¡ A ¡ x1 A x5 ¡ A x4 ¡ A ¡ Z A ¡ 1 Z A ¡ t Z A x3 x2Z ¡ ZA ¡ ZA 4x 3 t6 t

Fig. 2: A weighted complete graph with vertex set {1, 2, 3, 4}.

We have x1 +x2 +x3 = n/2 since this is the degree of vertex 1 in M(S). Hence x4 +x5 +x6 = n/2. We can conclude that the sum of weights of the edges of any star K1,3 and of any triangle K3 is

7 equal to n/2. In addition, we have x1 = x6, x2 = x5 and x3 = x4. Because of automorphisms it is sufficient to consider partitions of the vertex set of G determined by the weights x1, x2, x3 with 0 ≤ x1 ≤ x2 ≤ x3 ≤ n/2. We can now formulate an algorithm for the construction of the 163 graphs in the first layer by (0, 2)-extensions. For a given n we first construct all feasible triplets (x1, x2, x3). For each triplet (x1, x2, x3) we find in turn all ordered partitions of the vertex set of G into parts of cardinalities 0 x1, x2, . . . , x6. For each such partition we consider each relevant graph G and extend it to a graph G according to this partition. Then, for every graph G0 we calculate the least eigenvalue; if this is equal to −2 we have generated a regular exceptional graph on n + 4 vertices. Finally, we should eliminate isomorphic duplicates but record all successors of each graph G. A suitable computer program based on the above procedure has generated all 163 regular exceptional graph in the first layer. The 21 exceptional graphs from the second layer are constructed in a similar way, using (0, 2)-extensions of size 3. It turns out that the 3 graphs in the third layer can be constructed as successive (1, 3)-extensions of the line graph 2K2. The results are given in the Appendix of [19], where for each graph the list of its immediate successors is given. It is shown in [23], using the Hoffman bound, that the Petersen graph is the only non-Hamiltonian k-regular exceptional graph with a co-clique of size k + 1. It is conjectured that the Petersen graph is the only non-Hamiltonian strongly regular graph. The second topic of this section is the construction of 1-Salem graphs (defined below): these are L-graphs when non-bipartite, and they are used to construct certain algebraic integers. Recall that a Salem number is a real algebraic integer τ > 1 all of whose Galois conjugates 6= τ have modulus at most 1, with at least one having modulus exactly 1. Similarly, a Pisot number is a real algebraic integer θ > 1 whose Galois conjugates 6= θ have modulus strictly less than 1. Following [74]. we define a Salem graph as either (a) a non-bipartite graph having all eigenvalues but one in the closed interval [−2, +2], or (b) a bipartite graph having all eigenvalues but two in [−2, +2]. In what follows, we assume without loss of generality that Salem graphs are connected. Let λ be the largest eigenvalue of a Salem graph G. We set aside the ‘trivial’ cases in which (a) G is non-bipartite and λ ∈ Z, or (b) G is bipartite and λ2 ∈ Z. In all remaining cases, we obtain a Salem number as√ the larger√ solution of the equation z + 1/z = λ when G is non-bipartite, or as the larger solution of z + 1/ z = λ when G is bipartite (see [74, Proposition 3.1]). Let TG be the set of all Salem numbers constructed in this way. It is shown in [74, Theorem 1.1] that the set of limit points of TG is a set SG of Pisot numbers, and that the set TG ∪ SG is closed in IR: this supports the well-known conjecture of Boyd [137] that the set of all Salem and Pisot numbers is closed in IR. This technique for constructing Salem numbers from graphs was introduced by McKee et al [152], who began with starlike trees (trees with exactly one vertex of degree greater than 2). All the trees which define Salem numbers are described in [74]. Most of the subsequent results can be found in [55] and the PhD thesis of Gumbrell [11]. In the context of our book [4], we note that non-bipartite Salem graphs are L-graphs which are also reflexive, meaning that their second largest eigenvalue is at most 2. Consequently it is of interest to study reflexive generalized line graphs and reflexive exceptional graphs. In the remaining case, we must consider all reflexive bipartite graphs. Various classes of reflexive graphs have been studied in the literature (for example, trees [153], unicyclic graphs [83], bicyclic graphs [154] and cacti [82]). In principle only reflexive bipartite graphs need to be studied, as observed in [105], since the product of any non-bipartite Salem graph with K2 (as defined in [4, p.20]) is a bipartite Salem graph. However, it turns out that the non-biparitite case is easier to deal with, partly because the decomposition of bipartite graphs with respect to the above product need not be unique. An m-Salem graph is defined in [55] as a connected Salem graph for which m is the smallest number of vertices that must be deleted to obtain a graph whose components have index at most 2: thus these components are induced subgraphs of Smith graphs (as described in [4, Section 3.4]). Every Salem graph contains a 1-Salem graph as an induced subgraph, and so the 1-Salem graphs provide some necessary substructure. A connected bipartite graph G with index > 2 is a 1-Salem graph if and only if G has a vertex v such that each component of G − v is an induced subgraph of a Smith graph [55, Theorem 5]. The main result of [55] is a complete combinatorial description of all 1-Salem

8 graphs: the non-bipartite 1-Salem graphs are L-graphs, and here there are 25 infinite families and 383 sporadic examples. The infinite families and 6 of the sporadic graphs are generalized line graphs, while the remaining 377 sporadic graphs are exceptional graphs. The latter are found by computer, using representations in E8 (and exploiting the isometries of E8). The generalized line graphs are constructed from generalized cocktail party graphs (GCPs) in accordance with [4, Theorem 2.3.1]; the GCPs that arise, and the associated configurations, are severely restricted by the constraint on eigenvalues. Reflexive line graphs of trees are studied in [105]: it transpires that the trees which give rise to Salem graphs are induced subgraphs of certain ‘maximal trees’ (possibly infinite), and that these maximal trees exhibit only finitely many patterns (i.e. structures determined by certain integral parameters). The third topic of this section is the construction of connected non-bipartite integral graphs with index 3 [27]. By [5, Theorem 3.2.4], such a graph G has least eigenvalue −1 or −2. In the former case, G = K4 (see [5, Theorem 4.1.4]). In the latter case, G is either a generalized line graph or an exceptional graph, and it turns out that there are just 22 examples. We deal first with the generalized line graphs. Let G = L(H˜ ), where H˜ = H(a1, . . . , an) and H has degrees d1, . . . , dn. We define

Q(H˜ ) = A(H) + diag(d1 + 2a1, . . . , dn + 2an).

A vertex-edge matrix incidence matrix C for G is deﬁned in [4, p.6]: with appropriate orderings of vertices and edges, we have

Q(H˜ ) O C>C = A(L(H˜ )) + 2I,CC> = . O 2I

(In the case of line graphs, these relations reduce to those in Eq.(5), Section 10.) Since C>C and CC> have the same non-zero eigenvalues, it follows that if G is integral with index 3 then Q(H˜ ) has only integer eigenvalues and its spectral radius is 5. The authors of [27] identify the root graphs H˜ for which these two conditions are satisfied: the di, the ai and the eigenvalue multiplicities are severely constrained, and they find that 9 generalized line graphs arise Secondly, if G is an exceptional graph then G can be constructed as an extension of an exceptional star complement H for −2. There are 573 possibilities for H, and the 13 integral exceptional graphs G with index 3 are determined by computer: as described in Section 4, the first step is to find the cliques in each compatibility graph Γ(−2; H). We note that Salem graphs arise as induced subgraphs of these graphs. More generally, observe that reflexive integral graphs which are either non-bipartite L graphs or bipartite give rise to Salem graphs as induced subgraphs with index greater than 2. For example, one can inspect the 93 integral graphs with maximum degree 4. These were found by an exhaustive computer search (see [66]); in [14], one subclass of these graphs is examined without recourse to computers. Finally, see [131] for other constructions of integral graphs related to generalized line graphs.

7 Spectral characterizations and cospectral graphs

The papers [33, 34, 36] treat cospectral L-graphs by means of a computer and also theoretically. Here we present a selection of the results. Graphs with the same spectrum are called isospectral or cospectral graphs. The term “(unordered) pair of isospectral non–isomorphic graphs” will be denoted by PING. More generally, a “set of isospectral non–isomorphic graphs” is denoted by SING [35]. Thus a two element SING is a PING. A graph H which is cospectral with a graph G but not isomorphic to G, is called a cospectral mate of G. If {G1,G2,...,Gk} is a SING and if G is any connected graph, then the set {G1 ∪ G, G2 ∪ G, ...,Gk ∪ G} is also a SING. Each graph in the latter SING has a component isomorphic to a ﬁxed graph, namely G. A SING S is called reducible if each graph in S contains a component isomorphic to a ﬁxed graph; otherwise, S is called irreducible.

9 The table of cospectral graphs from [33, 36] contains irreducible SINGs in which graphs have least eigenvalue at least −2 and the number of vertices is at most 8. The table contains 201 irreducible L-SINGs with at most 8 vertices. This number includes 178 pairs, 20 triplets and 3 quadruplets of cospectral graphs. Although reducible SINGs should not be included in such tables, since they can easily be generated from irreducible ones, reducible SINGs are not without interest. For example, the (irreducible) SING No. 8.2 is a quadruple consisting of two (cospectral) reducible PINGs. (The ﬁrst and third graphs can be reduced to PING No. 6.2 while the other two reduce to the PING No. 7.2.) The unique PING with largest least eigenvalue and a minimal number of vertices is constructed in [35]. It is the PING No. 6.3 of the table and is reproduced in Fig. 3.

@@ @ r @ r r@ r @ @ r r r @ r @@ @ r r r r Fig. 3: An example of a PING.

Its least eigenvalue, approximately −1.5616, is the least solution of the equation λ2 − λ − 4 = 0. As usual, we deﬁne L1(H) = L(H) and Lk(H) = L(Lk−1(H)) for k = 2, 3,... . We can easily ﬁnd which L+–graphs are iterated line graphs Lk(H), k = 2, 3,... . Proposition 7.1. Let H be a connected graph and let L2(H) be an L+-graph. Then H is a path, or H is an odd cycle, or H consists of three nontrivial paths meeting at a vertex.

Proof. Let H1 = L(H) which implies G = L(H1). Then H1 is a tree or an odd unicyclic graph. In the ﬁrst case H has no vertices of degree greater then 2 and does not contain cycles. Hence, H = Pn. In the second case, if the odd cycle has length more than 3 then H is reduced to this cycle, hence H = C2k+1 (k ≥ 2). If H1 contains a triangle then either H = K3 or H consists of three nontrivial paths meeting at a vertex.

The following proposition is now straightforward. Proposition 7.2. Let k ≥ 3 and let Lk(H) be a connected graph in L+. Then H is a path or an odd cycle. Cospectral L–graphs could be line graphs, proper generalized line graphs and (generalized) exceptional graphs in all combinations. We shall consider here cospectrality of generalized line graphs with generalized exceptional graphs. For regular L–graphs the following theorem (see, for example, [4, Theorem 4.2.9]) is of great importance. Theorem 7.3. The spectrum of a graph G determines whether or not it is a regular connected line graph except for 17 cases. The exceptional cases are those in which G has the spectrum of L(H) where H is one of the 3-connected regular graphs on 8 vertices or H is a connected semi-regular bipartite graph on 6 + 3 vertices. It turns out that there are exactly 68 regular exceptional graphs which are cospectral with the 17 line graphs from Theorem 6.3. They are easily identiﬁed from the list of all 187 regular exceptional graphs given in Table A3 of [4]. Table A4 of [4] presents a construction of the these 68 graphs without recourse to a computer. We point out what Theorem 6.3 says in the case of iterated line graphs. Proposition 7.4. Let G = L2(H) where G is regular and H is a connected graph. If G has an exceptional cospectral mate then H is either K1,8 or K2,4. In the ﬁrst case the exceptional mates are the three Chang graphs, and in the second case they are graphs numbered 43-45 in [4, Table A3].

Proof. Let L(H) = H1. Then G = L(H1) and by Theorem 7.3 the graph H1 should be one of the 17 graphs appearing in that theorem. Only two of them are line graphs: the complete graph K8 and

10 the complement of the cube. In these cases we have H = K1,8 or H = K2,4. By consulting a table of regular exceptional graphs [4, Table A3] we can readily verify the last statement.

For non-regular graphs little is known; however, we can prove the following generalization of Theorem 7.3. Theorem 7.5. Let G be a generalized line graph L(H), where the B-graph H is connected. Then G has an exceptional graph G0 as a cospectral mate only if H satisﬁes one of the following conditions: (a) H has no petals and is bipartite of order at most 9, (b) H has no petals and is non-bipartite of order at most 8, (c) H has at least one petal, and the number of vertices after removing petals is at most 7. Proof. Let H have m edges and n vertices. By [4, Proposition 4.1.2], the multiplicity of −2 in G0 is m − p, where p ∈ {6, 7, 8}. The multiplicity k of −2 as an eigenvalue of L(H) is given by [4, Theorems 2.2.4, 2.2.8]. If H is a bipartite graph, then k = m − n + 1, and we have case (a). If H is a non-bipartite graph, then k = m − n and we have case (b). If G is a proper generalized line graph P then, in the notation of Section 3, k = m − n − i ai, and we have case (c).

Note that a connected proper GLG can have a disconnected generalized exceptional mate, as PING No. 8.10 shows for L+-graphs, and SING No. 8.44 for L0-graphs. Wang and coauthors studied spectral characterizations of several non-regular line graphs [121, 122, 123, 124]. A lollipop graph is obtained by identifying a vertex of a cycle with a pendant vertex of a path. Let g be the length of the cycle and n the number of vertices. It was claimed in [123] that line graphs of lollipop graphs were determined by their spectra, but an omission in the argument was noted in [123]; however the assertion holds for g ≥ 5 when g is odd and for n > 12 when g is even. A generalized cocktail-party graph of order at least 23 is characterized by its spectrum [121], as is the line graph of a double star [122]. In all of these characterizations the authors made use of an invariant of L-graphs called the star value S. This was introduced in [33], and it is deﬁned by

(−1)n S = P (n−k)(−2) = (λ + 2)(λ + 2) ··· (λ + 2), (n − k)! G 1 2 k where λ1, λ1, . . . , λk are all the eigenvalues of G greater than −2 (repetitions included). A double kite is a graph obtained from two complete graphs by connecting them by a path. Using [4, Theorem 2.3.20], Stani´c[111] shows that a double kite graph has no connected cospectral mate. Finally, it was proved in [133] that line graphs of starlike trees with maximum degree at least 12 are determined by their spectra.

8 The least eigenvalue: limit points and related graphs

We write λ(G) for the least eigenvalue of a graph G. The real number σ is a limit point for the least eigenvalue of a graph if there exists a sequence of distinct graphs G1,G2,... such that σ = limn→∞ λ(Gn). For example, −2 = limn→∞ λ(Pn). In what follows, ‘limit point’ means ‘limit point for the least eigenvalue of a graph’. As in Section 4, we write ∆(u) for the neighbourhood of a vertex u. It is well known that the least eigenvalue of any connected incomplete graph is less than −1. In the interval (−2, −1) there are inﬁnitely many numbers which are limit points for the least eigenvalue, and such numbers are described by the ﬁrst result below. If e is an edge of the graph G, we write Aˆ(G, e) for the matrix obtained from the adjacency matrix of (L(G)) by replacing the (e, e)-entry with −1. Note that Aˆ(G, e) = A(L(G)) − cc>, where c is an appropriate (0, 1)-vector. Theorem 8.1. The real number σ is a limit point greater than −2 if and only if σ is the least eigenvalue of Aˆ(T, e) for some pendant edge e of a tree T.

11 This result was proved by Hoffman [150] in 1977 with the qualification that λ(Aˆ(T, e)) < λ(L(T )), a relation that he conjectured was true of all pendant edges. Recently Greaves et al [53] proved Hoffman’s conjecture (with an extension to signed graphs); another proof is given in [79].√ This effectively deals with limit points greater than√ −2. The largest of these limit points is −τ = −( 5 + 1)/2 (see [161]), while the second largest is 3 (see [143]). An infinite family of line graphs of trees sharing the same least eigenvalue has been constructed in [80]: this family generalizes the family of coronas of complete graphs from [143]. In discussing limit points less than −2, we focus on the half-closed interval [−3, −2), where many limit points can be obtained by shifting the limit points from the interval [−2, −1) by −1. To see this it suffices to consider a sequence (Gn + K2), where limn→∞ λ(Gn) = θ ∈ [−2, −1). (Here the sum denotes a NEPS with basis {(0, 1), (1, 0)}; see [5]). Then θ − 1 is a limit point, but (as expected) there exist other limit points in [−3, −2) that cannot be obtained√ in this way. It was observed by Hoffman [151] that α = − 2 − 1 is the largest limit point less than −2. Subsequently, Woo and Neumaier [160] identified β ≈ −2.4812, the smallest root of the polynomial x3 + 2x2 − 2x − 2, as the second largest limit point less than −2. Two further limit points studied in the literature are −1 − τ (see [78]) and −3 (see [61]). Yu [129] provided infinitely many examples of connected regular graphs with least eigenvalue in the interval (β, −2). These results on limit points added impetus to the investigation of graphs whose least eigenvalue λ lies in the interval [−3, −2), and Hoffman graphs (defined below) have recently proved√ useful in this context. Hoffman had already proved in 1977 that there exists a function f :(−1− 2, −2] → IR such that any graph G with minimum degree δ(G) > f(λ) is a generalized line graph [151, Theorem 1.1]. A Hoffman graph H = (H; F,S) is a graph H together with a bipartition V (H) = F ∪˙ S, where F is an independent set of fat vertices and S is a dominating set of slim vertices. If also F is a dominating set then H itself is said to be fat. We write V (H) = V (H), E(H) = E(H), Vf (H) = F and 0 0 0 0 0 Vs(H) = S. An induced Hoffman subgraph of H is a Hoffman graph H = (H ; F ,S ) such that H is an induced subgraph of H with F 0 ⊆ F and S0 ⊆ S. One can regard a Hoffman graph as a graph constructed from an arbitrary graph G by adding independent vertices adjacent to V (G): the vertices in G are slim, and the new vertices are fat.

Example 8.2. For any graph G consider the Hoffman graph H obtained from G as follows: Vs(H) = V (G), and for each maximal clique K in G add a fat vertex adjacent to every vertex of K. Note that if |∆F (u) ∩ ∆F (v)| = 1 whenever u, v are adjacent vertices of G then G is a line graph (cf. [4, Theorem 2.1.1]): this notion will be generalized in due course. We can generalize Theorem 8.1 by introducing the reduced spectrum of a Hoffman graph H. The A C adjacency matrix of H may be written s , where A is the adjacency matrix of the slim C> O S subgraph of H (i.e. the subgraph induced by the slim vertices). The reduced spectrum of H is defined as the spectrum of the (symmetric) matrix

> B(H) = AS − CC .

Let λˆ(H) be the least eigenvalue of B(H). Then for any induced Hoffman subgraph H0 of H we have λˆ(H0) ≥ λˆ(H); in particular, if H0 is the slim subgraph of H, then λ(H0) ≥ λˆ(H). Theorem 8.3. (The Hoffman Limit Theorem; see [151, Proposition 3.1].) Let H be a Hoffman graph and let Hn be the graph obtained from H by replacing each fat vertex v by an n-clique Cv with the same neighbours in S as v. Then ˆ ˆ λ(Hn) ≥ λ(H) and limn→∞ λ(Hn) = λ(H). To generalize the notion of a line graph, we define a decomposition of a Hoffman graph H as follows. The family {Hi : i = 1, . . . , n} of non-empty induced subgraphs of the Hoffman graph H = (H; F,S) is said to be a decomposition of H if the following conditions hold:

n (i) V (H) = ∪i=1V (Hi);

12 (ii) Vs(Hi) ∩ Vs(Hj) = ∅ (1 ≤ i < j ≤ n);

(iii) if v ∈ Vs(Hi), w ∈ Vf (H) and v ∼ w then w ∈ Vf (Hi);

(iv) if u, v ∈ Vs(Hi)(u 6= v) then |∆F (u) ∩ ∆F (v)| = 1 when u ∼ v, and |∆F (u) ∩ ∆F (v)| = 0 when u 6∼ v. n In this situation we write H = ⊕i=1Hi. A Hoffman graph is decomposable if it has such a decomposition with n ≥ 2; otherwise, it is indecomposable. n If H = ⊕i=1Hi then by [61, Lemma 2.12] we have: ˆ ˆ λ(H) = min{λ(Hi) : 1 ≤ i ≤ n}. Accordingly one can focus attention on indecomposable Hoffman graphs. Now let F be family of Hoffman graphs. A graph is called an F-line graph if it is an induced n subgraph of the slim graph of some ⊕i=1Hi with each Hi ∈ F. For t ∈ IN, let H1,t be the Hoffman graph with one slim vertex and t fat vertices. Let H2,1 be the Hoffman graph with two non-adjacent slim vertices and one fat vertex. The smallest Hoffman graphs are illustrated in Fig. 4 in traditional form (where the larger vertices are the fat vertices).

LL LL r xL x rL r L L

Hx1,1 Hr1,2 Hx2,1 Fig. 4: The three smallest Hoﬀman graphs

Then the {H1,1}-line graphs are just the complete graphs, and the {H1,2}-line graphs are precisely the line graphs (Example 8.2). By [4, Theorem 2.3.1] the {H2,1, H1,2}-line graphs are the generalized line graphs. The {H2,1}-line graphs are the generalized cocktail party graphs; here there is just one fat vertex, and it is adjacent to every slim vertex. Now we can explain the role of Hoffman graphs in investigating graphs whose least eigenvalue is bounded below. In this situation, various induced subgraphs are forbidden by interlacing, and this restricts the√ configurations that can arise when joining slim vertices to fat vertices. (For example, if λ(H) > − t (t ∈ IN) then H has no induced K1,t.) Let H be a fat Hoffman graph with λ(H) ≥ r. If r = −1 then K1,2 is forbidden and H is complete; in particular the slim subgraph of H1 is complete, and there is just one fat vertex. If −2 ≤ r < −1 then each slim vertex is adjacent to just 1 or 2 fat vertices and the slim subgraph is an {H2,1, H1,2}-line graph, that is, a generalized line graph. √ When −3 ≤ r < −2 the picture is not yet complete, but for r ≥ −1 − 2 forbidden subgraphs ensure that the slim subgraph of H is an F-line graph for a specific family of size 4 [160]; for minimal forbidden subgraphs see [116, 117]. Analogous results for r ≥ −1 − τ can be found in [78] . When the fat indecomposable Hoffman graph H has a vertex with exactly two fat neighbours, necessary and sufficient structural conditions for λ(H) > −3 are known from [79, Theorem 4.3]. One consequence is the following result, which should be compared with Theorem 8.1. Theorem 8.4. (See [79, Theorem 5.2].) Let G be a graph with vertex v. If λ(Aˆ(G, v)) > −2 then G is the line graph of a tree T , and v corresponds to a pendant edge of T . A Hoffman graph H with λ(H) ≥ −2 is an L-graph, and so has a representation in one of the root systems An,Dn,E6,E7,E8 [4]. One aim of current research is to find analogous descriptions when −3 ≤ λ(H) < −2: here one can consider certain representations of slim subgraphs defined as follows. Let H be a Hoffman graph, and let m, n be positive integers. A reduced representation of H of norm n n m in IR is a function f : Vs(H) → R such that  m − |∆F (u)| if u = v, >  f(u) f(v) = 1 − |∆F (u) ∩ ∆F (v)| if u ∼ v,  −|∆F (u) ∩ ∆F (v)| if u 6∼ v.

13 Note that f(u)>f(v) is the (u, v) entry of B(H) + mI. Thus B(H) + mI is a Gram matrix, and so ˆ ˆ λmin(H) ≥ −m. Conversely, if λmin(H) ≥ −m then B(H) + mI is positive semi-deﬁnite and H has a n reduced representation of norm m. (Note that if H = ⊕i=1Hi then the images of the various Vs(Hi) are pairwise orthogonal.) If H = (H; F,S) then the special graph of H is the signed graph Hσ, with vertex set S, whose positive and negative edges are deﬁned by:

+ E (Hσ) = {uv : uv ∈ E(H), ∆F (u) ∩ ∆F (v) = ∅}, − E (Hσ) = {uv : uv 6∈ E(H), ∆F (u) ∩ ∆F (v) 6= ∅}.

By [61, Lemma 3.4], a Hoﬀman graph H is indecomposable if and only if the graph underlying Hσ is connected. If H is a fat Hoﬀman graph with λ(H) ≥ −3 then the negative subgraph of Hσ has a representation in one of the root systems (a) An or an extension thereof, (b) Dn or an extension thereof, (c) E6, E7 or E8. For this and further observations on representations of H, see [61]

9 The nullity of L-graphs

A graph G is singular if its adjacency matrix is singular, equivalently if 0 is an eigenvalue of G. The multiplicity of 0 is called the nullity of G, denoted by η(G). It was proved by Gutman and Sciriha [147] that the nullity of the line graph of any tree is at most 1. This result is equivalent to an observation of Grone et al in [146], that the multiplicity of 2 as a Laplacian eigenvalue of a tree is at most 1, a result obtained by using the coeﬃcient theorem for the characteristic polynomial of the Laplacian matrix. To establish the equivalence, recall that bipartite graphs have the same Laplacian and signless Laplacian spectrum, while for any connected graph H, the multiplicity of 0 in the adjacency spectrum of L(H) is the same as the multiplicity of 2 in the signless Laplacian spectrum of H (see Section 10). It was also observed in [147] that the nullity of line graphs of graphs other than trees can be arbitrarily large. A graph with a singular line graph is said to be L-singular. Such a graph T is heavy if a 0- eigenvector of L(T ) has no zero entries. Sciriha [156] observed that an L-singular tree has even order. In [73], it is shown that if the tree T is obtained from the disjoint union of two L-singular trees by adding a bridge then T is also L-singular. (The proof involves the straightforward construction of a 0-eigenvector of L(T ).) In the reverse direction, the authors consider modiﬁcations of a heavy L-singular tree T which result in smaller L-singular trees; when such a reduction is not possible, T is said to be a basic tree. A computer search revealed that there are only 11 basic L-singular trees among the 970 100 trees with 20 or fewer vertices. These initial observations were subsequently generalized in various directions. In [18] Bapat proved that η(L(G)) = 1 for any connected L-singular graph G with an odd number of spanning trees; moreover, every L-singular graph has even order. The authors of [67] show that if G is a unicyclic graph then η(L(G)) ≤ 2, and they distinguish the cases η(L(G)) = 0, 1, 2. All of the above observations are subsumed by the results of Zhou et al [134], who investigated + the nullity of L-graphs in general. For L -graphs, we have the following result, where En denotes the family of exceptional L+-graphs of order n ∈ {6, 7, 8}. Theorem 9.1. (See [134, Theorem 3.2].) Let G be a connected graph with least eigenvalue λ(G) > −2. Then the following hold: (i) if G = L(T ), where T is a tree, then η(G) ≤ 1;

(ii) if G = L(U), where U is an odd unicyclic graph, then η(G) ≤ 1; (iii) G = L(T ; 1, 0,..., 0), where T is a tree, then 1 ≤ η(G) ≤ 2;

(iv) if G ∈ E6 ∪ E8 then η(G) = 0;

(v) if G ∈ E7 then η(G) ≤ 1.

14 In case (iii) let T˜ be the tree obtained from G by deleting one edge of the only petal. Then η(L(G)) = 2 if and only if T˜ is L-singular. The 26 L-singular graphs that arise in case (v) of Theorem 9.1 are listed in [134, Theorem 3.3], found by inspecting the 110 graphs in E7 [4, Table A2]. It remains an open problem to determine the exceptional graphs with nullity greater than 1. In the case of L0-graphs, we have the following result for line graphs.

Theorem 9.2. (See [134, Theorem 3.5].) Let G be a graph with cyclomatic number k. If λ(L(G)) = −2 then η(L(G)) ≤ k + 1,; moreover, when η(L(G)) = k + 1 we have: (i) every spanning tree of G is L-singular;

(ii) for any edge of G other than a bridge, η(G − e) = k; (iii) G is bipartite of even order.

Pn Next we consider proper generalized line graphs. Let Ga = G(a1, a2. . . . , an), where i=1 ai > 0. − Let Ga be the graph obtained from Ga by deleting one edge from each petal. Then we have: Theorem 9.3. (See [67, Theorem 3.10].) With the above notation,

n − X η(L(Ga)) = η(L(Ga ) + ai. i=1

Here the second summand arises from the petals as duplicate vertices in L(Ga). It was conjectured in [134] that the girth of a unicyclic graph U is divisible by 4 if η(L(U)) = 2: this conjecture has now been conﬁrmed in [52, Corollary 17]. In [52] Ghorbani considers even integers as possible Laplacian or signless Laplacian eigenvalues of an arbitrary connected graph G. He explains how the the multiplicity of such eigenvalues is controlled by the number of spanning trees, here denoted by τ(G). As a by-product one can deduce the same type of results for the adjacency spectrum of L(G). For example, if t is the largest integer such that 2t divides τ(G), then the multiplicity of any even integer 6= −2 as an eigenvalue of A(L(G)) is at most t + 1 [52, Theorem 1]. Moreover, in the case that G has odd order and τ(G) is not divisible by 4 we have: (i) every even integer eigenvalue µ of L(G) is congruent to 2 mod 4, (ii) there is at most one such eigenvalue µ 6= −2, and then µ is simple [52, Theorem 2].

10 A spectral theory of graphs based on the signless Laplacian

Recall that, given a graph G, the matrix Q = D + A is called the signless Laplacian of G, where A is the adjacency matrix of G and D is the diagonal matrix of vertex degrees. The matrix L = D − A is well-known as the Laplacian of G. Recently, a spectral theory of graphs based on the signless Laplacian has been developed (using the signless Laplacian without explicit involvement of other graph matrices) [31, 38, 42]. To motivate the study of Q-theory we present some relevant computational results. The papers [144], [58] provide the following spectral uncertainties for the sets of all graphs of order n ≤ 11. Here rn, sn and qn are the spectral uncertainties with respect to A, L and Q respectively.

n 4 5 6 7 8 9 10 11 rn 0 0.059 0.064 0.105 0.139 0.186 0.213 0.211 sn 0 0 0.026 0.125 0.143 0.155 0.118 0.090 qn 0.182 0.118 0.103 0.098 0.097 0.069 0.053 0.038

15 We see that the numbers qn are smaller than the numbers rn and sn for n ≥ 7. In addition, the sequence qn is decreasing for n ≤ 11 while the sequence rn is increasing for n ≤ 10. This is a strong basis for believing that studying graphs by means of their Q–spectra is more eﬃcient than studying their (adjacency) spectra. Moreover, a connection with the spectra of line graphs and the existence of a well developed theory of graphs with least eigenvalue −2 [4] were used as additional arguments for studying eigenvalues of the signless Laplacian. To describe the connection with line graphs, let R be the vertex-edge incidence matrix of a graph G. The following relations are well known:

RRT = A + D,RT R = A(L(G)) + 2I, (5) where A(L(G)) is the adjacency matrix of L(G). If G has m edges and n vertices then from relations (5) we obtain m−n PL(G)(x) = (x + 2) QG(x + 2), (6) where QG(x) is the characteristic polynomial of the signless Laplacian. The polynomial QG(x) will be called the Q-polynomial of the graph G. The signless Laplacian is a positive semideﬁnite matrix, and so all its eigenvalues are non-negative. Concerning the least eigenvalue we have the following observation from [31]. Proposition 10.1. The least eigenvalue of the signless Laplacian of a connected graph is equal to 0 if and only if the graph is bipartite. In this case 0 is a simple eigenvalue. Proof. By [4, Theorem 2.2.4 ] the multiplicity of the eigenvalue −2 in L(G) is m − n + 1 if G is bipartite, and m − n if G is not bipartite. This together with formula (6) yields the assertion of the proposition.

Corollary 10.2. In any graph the multiplicity of eigenvalue 0 of the signless Laplacian is equal to the number of bipartite components. A formula giving a structural interpretation of the coeﬃcients of the Q-polynomial was proved in [38] using A-polynomials of line graphs. The problem of graph reconstruction from collections of subgraphs of various kinds is a traditional challenge in graph theory. The next result, from [42], involves edge-deleted subgraphs. Theorem 10.3. The Q–polynomial of a graph G is reconstructible from the collection of the Q– polynomials of edge-deleted subgraphs of G. Proof. Given the Q–polynomials of edge-deleted subgraphs of G, we can use the formula (6) to calculate the A–polynomials of vertex-deleted subgraphs of the line graph L(G). As is well known, the A-eigenvalues of line graphs are bounded from below by −2. By the results of [102] the A– polynomial of L(G) can now be reconstructed. Using formula (6) again, we obtain the Q–polynomial of G. The reconstruction of the Q–polynomial of a graph G from the collection of the Q–polynomials of edge-deleted subgraphs of G corresponds to the reconstruction of the A-polynomial of a graph G from the collection of the A-polynomials of the vertex-deleted subgraphs of G. While the ﬁrst problem is positively solved by Theorem 10.3, the corresponding problem in A-theory remains unsolved in the general case. Therefore Theorem 10.3 tells us much about the usefulness of Q–theory. The Q-polynomial of a semi-regular bipartite graph was obtained in [43] using [4, Proposition 1.2.18]. Theorem 5.2.2 of [4] was used in [25] to identify the graph whose least Q-eigenvalue is smallest among connected non-bipartite graphs with a prescribed number of vertices. Some Q-integral graphs are constructed in [103, 104] and [109]. Many other results on the signless Laplacian have been obtained since the publication of [31, 38] and [39]: see the bibliographies in [42] and [43]. The number of papers on the signless Laplacian is about 200 while the article [38] has been cited in about 170 papers.

16 11 Signed graphs

Here, as in [4, Section 7.8], we regard a signed graph (or sigraph) as a pair Gσ = (G, σ) consisting of a simple graph G = (V,E) and a signature map σ : E → {+1, −1}. Thus E = E+ ∪ E− where + −1 − −1 E = σ (+1) and E = σ (−1). The adjacency matrix of Gσ is the matrix Aσ = (aij), where aij = σ(ij) if i ∼ j and aij = 0 otherwise. We write λ(Gσ) for the least eigenvalue of Aσ. The sigraphs 0 0 Gσ and Gσ0 are isomorphic if there exists a signature-preserving graph isomorphism φ : G → G (so that σ(vw) = σ0(φ(v)φ(w))). An orientation of Gσ is a map η : V × E → {−1, +1, 0} such that (i) η(u, vw) = 0 whenever u 6= v, w, (ii) η(v, vw)η(w, vw) = −σ(vw). Now we can define the vertex-edge incidence matrix Bη of a sigraph with orientation η as the matrix with (v, e)-entry η(v, e). It follows that, for any orientation η, we have > BηBη = D − Aσ, (7) where D is the diagonal matrix of vertex degrees of G. The matrix D − Aσ is called the Kirchhoff + − matrix of Gσ: when E = E it is the Laplacian matrix of G, and when E = E it is the signless Laplacian matrix of G. If multiple edges are allowed (as they are in the root graphs of generalized line graphs) then Eq.(7) remains valid, with aij equal to the difference between the number of positive edges and the number of negative edges between i and j. If the sigraph Gσ has an orientation η then we can define a line sigraph L(Gσ) as the graph L(G) with signature σL given by: σL(eiej) = η(v, ei)η(v, ej), where ei ∩ ej = {v}. It follows that

T Bη Bη = 2I + AσL (L(Gσ)). (8)

Here the construction of L(Gσ) depends on η, but diﬀerent orientations of Gσ result in line digraphs that are switching-equivalent in accordance with the following deﬁnition. First, switching at the vertex v is the interchange of signs on the edges containing v. Then the 0 0 sigraphs Gσ and Gσ0 are switching-equivalent if there exists W ⊆ V (G) such that Gσ0 is isomorphic to the sigraph obtained from Gσ by switching at each vertex of W . Note that switching-equivalence is an equivalence relation, and switching-equivalent graphs have similar adjacency matrices. Moreover, switching-equivalent graphs can be oriented so that their line sigraphs are isomorphic. From Eq.(8) we see that the least eigenvalue of any line sigraph is greater than or equal to −2. A sigraph Gσ with λ(Gσ) ≥ −2 is not necessarily a line sigraph, but from [138] we have: Theorem 11.1. Any signed graph with least eigenvalue greater than or equal to −2 has a representation in a root system Dn, for some n, or E8.

If the connected sigraph Gσ is representable in some Dn then Aσ + 2I is the Gram matrix of n vectors v1,..., vm in IR such that every vi has just two non-zero entries, each ±1. Thus [v1| · · · |vm] can be regarded as the oriented incidence matrix of a sigraph Gσ, possibly with multiple edges. In view of Eq.(8), Gσ is the root sigraph of a line sigraph, and we shall describe the root sigraphs that arise when λ(L(Gσ)) > −2. Note that if G = L(U), where U is a unicyclic graph of girth > 3 then each edge e1e2 of G lies in a unique maximal clique of G, here denoted by K(e1e2). Now we can state the following result (see [53, Corollary 3 and Theorem 6]).

Theorem 11.2. Let Gσ be a connected signed graph with least eigenvalue greater than −2. If Gσ is representable in Dn for some n, then G = L(H) where H is one of the following graphs, with E(L(H)) = E+ ∪ E−: (i) H is a tree or a unicyclic with an odd cycle, and E+ = E(L(H));

− (ii) H is unicyclic with an even cycle C, E = {uv ∈ E(L(H)) : v ∈ K(e1e2)}, where e1e2 is an edge of L(C);

(iii) H is a tree with one pair of duplicate edges e, e0, and E+ consists of E(L(H − e0)) together with edges between e0 and all vertices 6= e in a a maximal clique of L(H − e0) containing e.

17 Conversely, if Gσ is a sigraph described by (i), (ii), or (iii) above, then Gσ is representable in Dn for some n and λ(Gσ) ≤ −2.

We say that a sigraph Gσ with λ(Gσ) ≥ −2 is exceptional if it is not representable in some Dn. Those with λ(Gσ) > −2 were determined by the authors of [53], using representations in E8. They are listed in the Appendix to [53], and we can summarize the results as follows.

Theorem 11.3. Let Gσ be an exceptional sigraph with least eigenvalue greater than −2. Then Gσ is one of (i) 32 sigraphs on 6 vertices, (ii) 233 sigraphs on 7 vertices, (iii) 1242 sigraphs on 8 vertices. Theorems 11.2 and 11.3 together generalize a result of Doob and Cvetković[145] which classifies the L+-graphs [4, Theorem 2.3.20]. Finally, we note that many of the results established for finite graphs or sigraphs have been generalized to infinite graphs or sigraphs. The interested reader is referred to earlier papers by Torgaˇsev[157, 158], and to those of Vijayakumar et al [118, 119, 120, 140, 159], some of which are recent.

12 Computer Science and Control Theory

The papers [44, 45] and [17] provide a survey of applications of the theory of graph spectra to Computer Science. All three papers describe applications to statistical databases of results concerning the eigenspace of eigenvalue −2 in generalized line graphs, as already mentioned in [4, pp.128-129]. In fact, this application was noted in [BrCv], where further tools for treating statistical databases are developed. The paper [44] gives some details including bibliographical data. Let L(H) be a generalized line graph with least eigenvalue −2. An edge u of H is called weak if the coordinate corresponding to u in each −2-eigenvector of L(H) is equal to 0. Edges of H which are not weak are called strong, and the subgraph of H induced by the strong edges is called the L-core of H. The next two propositions are straightforward. Recall that an odd dumb-bell consists of two odd cycles joined by a path, possibly of zero length, but otherwise disjoint. Proposition 12.1. The L-core of a graph H is empty if and only if H contains neither even cycles nor odd dumb-bells.

Proposition 12.2. The L-core of a graph is empty if and only if L(H) has least eigenvalue greater than −2. The connected graphs with an empty L-core are described in [4, Theorem 2.3.20]. The main observation is given by the following theorem. Recall that the angles of a graph associated with an eigenvalue λ are the cosines of the angles between E(λ) and the co-ordinate axes. Theorem 12.3. The L-core of a graph H is induced by the edges of H which correspond to the non-zero angles for the eigenvalue −2 in L(H). A non-spectral algorithm for ﬁnding the L-core of a graph is described in [BrCv]. The importance of the L-core of a graph can be seen from the following reformulation of Theorem 4 from [29].

Theorem 12.4. Let H1 be the L-core of a graph H. Then the eigenspace of the eigenvalue −2 in L(H1) is the projection of the eigenspace of the eigenvalue −2 in L(H) to the subspace corresponding to strong edges of H. Theorem 12.4 says that the two eigenspaces are identical if we neglect zero coordinates in the eigenvectors of H corresponding to weak edges of H. Thus the eigenspace of −2 for H is determined by the L-core of H. From a computational point of view this means a reduction in dimension for the problem of ﬁnding a basis for E(−2). The additional possibility that the L-core is not connected allows a further reduction in dimension since each component of the L-core can be treated separately.

18 Finally, the paper [40] is related to Control Theory. An eigenvalue of a graph is called main if the corresponding eigenspace contains a vector for which the sum of coordinates is diﬀerent from 0. The connected graphs in which all eigenvalues are simple and main are called controllable, in view of some applications in control theory. Some general characterizations of controllable L-graphs are given; in particular, all the controllable exceptional graphs are completely determined. The controllable graphs whose second largest eigenvalue does not exceed 1 are also studied.

13 Miscellaneous Pn Recall that the energy of a graph with eigenvakues λ1, . . . , λn is defined as i=1 |λi|. The paper [32] studies the graphs with maximal energy for a prescribed number of vertices. It appears that graphs with least eigenvalue −2, particularly exceptional graphs, are in many cases graphs with maximal energy. This fact can perhaps be explained by the high multiplicity of −2, which is in accordance with the main conclusion of [32] that maximal energy graphs should have a small number of distinct eigenvalues. The energy and some other spectral invariants of certain L-graphs are discussed in [56, 65, 72, 75, 84, 85, 86, 132] in the context of Chemistry. In particular, the paper [132] deals with the eigenvalues of the distance matrix D˜ of a graph and with the corresponding energy. It is shown that a regular graph of diameter 2 has exactly one positive D˜-eigenvalue if and only if the least A-eigenvalue is at least −2. Using the tables from [4]. the authors find 7 strongly regular exceptional graphs; they remark that there may be other exceptional graphs with diameter 2. n+1 When n is even, let H = n/2 and L = H + 1; when n is odd, let H = L = 2 . The eigenvalues λH and λL feature in chemical graph theory in the study of molecular stability, and the HL-index of G is defined as R(G) = max {|λH |, |λL|}. There exist graphs with arbitrarily large HL-index [60]. It is shown in [77] using Theorem 2.3.20 and√ Corollary 2.3.22 of [4] that if G is a connected bipartite graph with ∆(G) = ∆ ≥ 3 then R(G) ≤ ∆ − 2 unless G is the incidence graph of a projective plane of order ∆ − 1. The structure of the eigenspace of the eigenvalue −2 in generalized line graphs and in exceptional graphs is studied in [29]. Here we start with some elementary observations related to integral eigenvalues of a graph. Proposition 13.1. The eigenspace of an integral eigenvalue of a graph has a basis consisting of vectors with integer coordinates. A vector is called integral if all its coordinates are integers. An integral vector v 6= 0 is called minimal if it has the smallest possible norm among all non-zero scalar multiples of v. A basis of a vector space is called integral (minimal integral) if it consists of integral (minimal integral) vectors. In view of Proposition 12.1 we have:

Proposition 13.2. The eigenspace of any integral eigenvalue of any graph has at least one minimal integral basis. From [4, Remark 5.2.11] we know that in generalized line graphs with least eigenvalue −2, the eigenspace of −2 always has at least one minimal integral basis in which the vectors have coordinates of modulus at most 2. However, data from the literature show that in exceptional graphs, vectors with coordinates of larger modulus can appear. Therefore the following definitions are appropriate. The height of a vector is the maximal modulus of its coordinates. The height of a basis is the maximal height of its vectors. We say that an L0-graph is k-degenerate if k is the minimal height of a (minimal) integral basis of the eigenspace of −2. This question of degeneracy has been considered in reference [Haz] of [4] (see also [4, p.129]) in relation to the so-called Hodge cycles on certain abelian varieties of CM-type. The above definition of the k-degeneracy is a slight modification of the definition given in [Haz]. Further details appear in [37], while the following result appears as Theorem 8 of [29].

19 Theorem 13.3. The degeneracy of exceptional graphs having the graph H440 as a star complement for the eigenvalue −2 is at most 2. Since the graph H440 is an almost universal star complement for generating maximal exceptional graphs, Theorem 13.3 covers most of the exceptional graphs. Nevertheless several exceptional graphs have higher degeneracies – up to 6, as examples and computations show (cf. the Smith graphs in [4, Fig. 1]). A graph with 9 vertices and degeneracy 5 is given in Fig. 5. −3 H H 1 r @ ¨r 1 1r− r2 3 r− r4 5 r @¨ r −r3 Fig. 5: A graph of degeneracy 5.

If A is the adjacency matrix of an L-graph G of order n then A + 2I is the Gram matrix of n vectors in the Euclidean space IRt, where t is the comultiplcity of −2 as an eigenvalue of G. These n vectors determine a system of lines in at angles of 60o and 90o in IRt. The classification of such line systems by Cameron et al [138], and its implications for L-graphs, is described in [4, Chapter 3]. The extensions of these line systems to a complex space is described in [64] and [7, Section 7.10], based on the treatment in [4]: the context is the classification of the unitary reflection groups. See also [6]. The chromatic number χ(G) of a graph G is easily seen to be at least the clique number ω(G) and at ω(G)+∆(G)+1 most one more than the maximum degree ∆(G). Reed [155] conjectured that χ(G) ≤ d 2 e. It is shown in [51] that this conjecture holds for generalized line graphs and exceptional graphs with least eigenvalue greater than −2. Recall that a vertex v of the graph G is a downer (neutral, Parter) vertex with respect to an eigenvalue µ of multiplicity k > 0 if its multiplicity in G − v is k − 1 (resp. k, k + 1). For µ = −2, the vertices of line graphs are classified accordingly in [101]. The occurrence of repeated eigenvalues in the line graph of a tree is considered in [100], in the context of the polynomial reconstruction problem [4, Section 7.5]. A local graph inside a graph G is a subgraph induced by the neighbours of a vertex in G. The spectra of local graphs in line graphs are discussed in [48, 49], while the authors of [63] investigate distance–regular graphs in which λ2(H) ≤ 1 for every local graph H. Finally, let S(G) be the subdivision graph of a graph G (obtained from G by inserting a vertex in m−n 2 each edge). If G has n vertices and m edges then PS(G)(x) = x QG(x ) (see [2, p.63]). At the m−n same time we have PL(G)(x) = x QG(x + 2) from Eq.(6). Connections between the eigenspaces of A(S(G)), A(L(G)) and the signless Laplacian of G (for appropriate eigenvalues) are established in [99].

References

(a) Completion of incomplete references appearing in the book [BaSi3] K. T. Balińska, S. K. Simić,K. T. Zwierzyński,Which nonregular, bipartite, integral graphs with maximum degree four do not have ±1 as eigenvalues?, Discrete Math. 286 (2004) 15-25. [BLMS] F. K. Bell, E. M. Li Marzi, S K. Simić,Some new results on graphs with least eigenvalue not less than −2, Rendiconti del Seminario Matematica di Messina, Ser. II 9 (2003) 11-30. [BrCv] Lj. Branković,D. Cvetković,The eigenspace of the eigenvalue −2 in generalized line graphs and a problem in security of statistical data bases, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 14 (2003) 37-48.

20 [CvRS7] D. Cvetković,P. Rowlinson, S. Simić,Graphs with least eigenvalue −2; a new proof of the 31 forbidden subgraphs theorem, Designs, Codes and Cryptography 34 (2005) 229-240. [Row4] P. Rowlinson, Star complements and the maximal exceptional graphs, Publ. Inst. Math. (Beograd) 76(90) (2004) 25-30. [Ste1] D. Stevanović,4-Regular integral graphs avoiding ±3 in the spectrum, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 14 (2003) 99-110.

(b) Books [1] A. E. Brouwer, W. H. Haemers, Spectra of graphs, Springer, New York, 2011. [2] D. Cvetković,M. Doob, H. Sachs, Spectra of Graphs; Theory and Application, third edition, Johann Ambrosius Barth Verlag, Heidelberg-Leipzig, 1995. [3] D. Cvetković,P. Rowlinson, S. Simić,Eigenspaces of Graphs, Cambridge University Press, Cambridge, 1997. [4] D. Cvetković, P. Rowlinson, S. Simić,Spectral Generalizations of Line Graphs: On Graphs with Least Eigenvalue −2, Cambridge University Press, Cambridge, 2004. [5] D. Cvetković,P. Rowlinson, S. Simić,An Introduction to the Theory of Graph Spectra, Cambridge University Press, Cambridge, 2009. [6] J. Jost, Riemannian Geometry and Geometric Analysis, Springer, Heidelberg, 2011. [7] G. I. Lehrer, D. E. Taylor, Unitary reflection groups, Cambridge University Press, Cam- bridge, 2009. [8] P. van Mieghem, Graph Spectra for Complex Networks, Cambridge University Press, Cam- bridge, 2011. [9] R. Stekolshchik, Notes on Coxeter Transformations and the McKay Correspondence, Springer, Berlin, 2008.

(c) Doctoral theses [10] Z. Stani´c,Some reconstructions in spectral graph theory and graphs with integral Q-spectrum (in Serbian), PhD thesis, Faculty of Mathematics, University of Belgrade, 2007. [11] L. Gumbrell, On spectral constructions for Salem graphs, PhD Thesis, Royal Holloway, University of London, 2013. [12] T. Koledin T, Some classes of spectrally constrained graphs (in Serbian), PhD thesis, Faculty of Mathematics, University of Belgrade, 2013

(d) New papers [13] N. M. M. de Abreu, Old and new results on algebraic connectivity of graphs, Linear Algebra Appl. 423 (2007) 53-73. [14] N. M. M. de Abreu, K. T. Bali´nska, S. K.‘Simi´c,More on non-regular bipartite graphs with maximum degree four not having ±1 as eigenvalues, Appl. Anal. Discrete Math. 8 (2014) 123-154. [15] S. Akbari, E. Ghorbani, A. Mahmoodi, On graphs whose star sets are (co-)cliques, Linear Algebra Appl. 430 (2009) 504-510.

21 [16] M. Andeli´c,D. M. Cardoso, S. K. Simi´c,Relations between (κ, τ)-regular sets and star complements, Czechoslovak Math. J. 63(138) (2013) 73-90.

[17] B. Arsić,D. Cvetković,S. K. Simić,M. Skarić,Graphˇ spectral techniques in computer sciences, Appl. Anal. Discrete Math. 6 (2012) 1-30. [18] R. B. Bapat, A note on singular line graphs, Bull. Kerala Math. Assoc. 8 (2011) 207-209. [19] I. Barbedo, D. M. Cardoso, D. Cvetković,P. Rama, S. K. Simić,A recursive construction of regular exceptional graphs with least eigenvalue −2, Portugal Math., to appear.

[20] F. K. Bell, D. Cvetkovi´c,P. Rowlinson, S. K. Simi´c,Graphs for which the least eigenvalue is minimal I, Linear Algebra Appl. 429 (2008) 234-241. [21] N. L. Biggs, Some properties of strongly regular graphs, arXiv: 1106.0889v1 (June 2011).

[22] V. Brankov, D. Cvetković,S. K. Simić,D. Stevanović,Simultaneous editing and multil- abelling of graphs in system newGRAPH, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 17 (2006) 112-121. [23] A. E. Brouwer, W. H. Haemers, Hamiltonian strongly regular graphs, http://ssrn.com/abstract=1104963. [24] D. M. Cardoso, D. Cvetković,Graphs with least eigenvalue −2 attaining a convex quadratic upper bound for the stability number, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math. 133 (2006) 42-55. [25] D. M, Cardoso, D. Cvetković,P. Rowlinson, S. K. Simić,A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph, Linear Algebra Appl. 429 (2008) 2770-2780. [26] D. M. Cardoso, M. Kaminski, V. Lozin, Maximum k-regular induced subgraphs, J. Comb. Optim. 14(4) (2007) 455-463. [27] T. Chung, J. Koolen, Y. Sano, T. Taniguchi, The non-bipartite integral graphs with spectral radius three, Linear Algebra Appl. 435 (2011) 2544-2559.

[28] T.-C. Chang, B.-S. Tam, S.-H. Wu, Theorems on partitioned matrices revisited and their applications to graph spectra, Linear Algebra Appl. 434 (2011) 559-581. [29] D. Cvetkovi´c, Graphs with least eigenvalue −2: the eigenspace of the eigenvalue −2, Rendi- conti Sem. Mat. Messina, Ser. II 25(9) (2003) 63-86.

[30] D. Cvetkovi´c,Notes on maximal exceptional graphs, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 15 (2004) 103-107. [31] D. Cvetkovi´c,Signless Laplacians and line graphs, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math. 131 (2005) 85-92.

[32] D. Cvetković,J. Grout, Maximal energy graphs should have a small number of distinct eigenvalues, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math., 134 (2007) 43-57. [33] D. Cvetković,M. Lepović,Sets of cospectral graphs with least eigenvalue at least −2 and some related results, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math. 129 (2004), 85-102.

[34] D. Cvetkovi´c,M. Lepovi´c,Cospectral graphs with least eigenvalue at least −2, Publ. Inst. Math. (Beograd) 78(92) (2005), 51-63.

22 [35] D. Cvetkovi´c,M. Lepovi´cM., Towards an algebra of SINGs, Univ. Beograd, Publ. Elek- trotehn. Fak., Ser. Mat. 16 (2005) 110-118.

[36] D. Cvetković,M. Lepović,A table of cospectral graphs with least eigenvalue at least −2, in: D. Cvetković,Iracionalno u racionalnom, Akademska misao, Beograd, 2011, pp.173-196; see also http://www.mi.sanu.ac.rs/projects.htm/results1389.htm. [37] D. Cvetković,P. Rowlinson, S. K. Simić,Star complements and exceptional graphs, Linear Algebra Appl. 423 (2007) 146-154.

[38] D. Cvetković,P. Rowlinson, S. K. Simić,Signless Laplacians of finite graphs, Linear Algebra Appl. 423 (2007) 155-171. [39] D. Cvetković,P. Rowlinson, S. K. Simić,Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math. (Beograd) 81(95) (2007) 11-27.

[40] D. Cvetković,P. Rowlinson, Z. Stanić,M,-G. Yoon, Controllable graphs with least eigenvalue at least −2, Appl. Anal. Discrete Math. 5 (2011) 165-175. [41] D. Cvetković,S. K. Simić,Errata, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 15 (2004) 112. [42] D. Cvetković,S. K. Simić,Towards a spectral theory of graphs based on the signless Laplacian I, Publ. Inst. Math. (Beograd), 85(99) (2009) 19-33. [43] D. Cvetković,S. K. Simić,Towards a spectral theory of graphs based on the signless Laplacian II, Linear Algebra Appl. 432 (2010) 2257-2272. [44] D. Cvetković,S. K. Simić,Graph spectra in computer science, Linear Algebra Appl. 434 (2011) 1545-1562. [45] D. Cvetković, S. K. Simić,Spectral graph theory in computer science, The IPSI BgD Trans- actions on Advanced Research 8 (2012) 35-42. [46] A. Erić,C. M. da Fonseca, Some consequences of an inequality on the spectral multiplicity of graphs, Filomat 27 (2013) 1455-1461.

[47] Y.-Z. Fan, F.-F. Zhang, Y. Wang, The least eigenvalue of the complements of trees, Linear Algebra Appl. 435 (2011) 2150-2155. [48] M. A. Fiol, M. Mitjana, The local spectra of line graphs, 6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, Electron. Notes Discrete Math. 28, Elsevier Sci. B. V., Amsterdam (2007) 95-102. [49] M. A. Fiol, M. Mitjana, The local spectra of regular line graphs, Discrete Math. 310 (2010) 511-517. [50] G. K. Fricke, B. W. Rogers, D. P. Garg, On the stability of swarm consensus under noisy control, arXiv:1010.4999 (2010).

[51] D. Gernert, L. Rabern, A computerized system for graph theory, illustrated by partial proofs for graph-coloring problems, Graph Theory Notes of New York 22 (2008) 14-24. [52] E. Ghorbani, Spanning trees and even integer eigenvalues of graphs, Discrete Math. 324 (2014) 62-67.

[53] G. Greaves, J. Koolen, A. Munemasa, Y. Sano, T. Taniguchi, Edge-signed graphs with smallest eigenvalue greater than −2, J. Comb.Theory Ser. B, DOI: 10.1016/j.jctb.2014.07.006

23 [54] L. Gumbrell, A connection between the bipartite complements of line graphs and the line graphs with two positive eigenvalues, arXiv:1204.2981 (2012). [55] L. Gumbrell, J. McKee, A classification of all 1-Salem graphs, to appear. [56] I. Gutman, M. Robbiano, E. A. Martins, D. M. Cardoso, L. Medina, O. Rojo, Energy of line graphs, Linear Algebra Appl. 433 (2010) 1312-1323. [57] W. H. Haemers, Matrices and Graphs, CentER Discussion Paper No. 2005-37, Tilburg University (2005), available at SSRN: http://ssrn.com/abstract=706843 or http://dx.doi.org/10.2139/ssrn.706843. [58] W. Haemers, E. Spence, Enumeration of cospectral graphs, Europ. J. Comb. 25 (2004) 199- 211. [59] L.-H. Huang, B.-S. Tam, S.-H. Wu, Graphs whose adjacency matrices have rank equal to the number of distinct nonzero rows, Linear Algebra Appl. 438 (2013) 4008-4040. [60] G. Jakliˇc,P. W. Fowler, T. Pisanski, The HL-index of a graph, Ars Math. Contemp. 5 (2012) 99-105. [61] H. J. Jang, J. Koolen, A. Munemasa, T. Taniguchi, On fat Hoffman graphs with smallest eigenvalue at least −3, Ars Math. Contemp. 7 (2014) 105-121. [62] T. Koledin, Z. Stanić,Regular graphs whose second largest eigenvalue is at most 1, Novi Sad J. Math., 43(3) (2013) 145-153. [63] J. H. Koolen, H. Yu, The distance-regular graphs such that all of its second largest local eigenvalues are at most one, Linear Algebra Appl., 435(2011), 2507-2519. [64] M. Krishnasamy, D. E. Taylor, Embeddings of complex line systems and finite reflection groups, J. Australian Math. Soc., 85(2008) 211-228. [65] W. Lang, L. Wang, Energy of generalized line graphs, Linear Algebra Appl., 437(2012), no. 9, 2386-2396. [66] M. Lepović,S. K. Simić,K. T. Balińska, K. T. Zwierzyński,There are 93 non-regular, bipartite integral graphs with maximum degree four, The Technical University of Poznań, CSC Report No. 511, Poznań,2005. [67] H.-H. Li, Y.-Z. Fan, L. Su, On the nullity of the line graph of unicyclic graph with depth one, Linear Algebra Appl. 437 (2012) 2038-2055. [68] S. Li, L. Zhang, Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs, Linear Multilinear Algebra 59 (2011) 145-158. [69] S. Li, L. Zhang, Permanental Bounds for the signless Laplacian matrix of a unicyclic graph with diameter d, Graphs and Combinatorics 28 (2012) 531-546. [70] D. Liu, S. Trajanovski, P. Van Mieghem, Random line graphs and a linear law for assorta- tivity, Phys. Rev. E 87, 012816, 31 January 2013. [71] D. Liu, S. Trajanovski, P. Van Mieghem, ILIGRA: An efficient inverse line graph algorithm, J. Math. Model Algor., to appear. [72] Z. Liu, Energy, Laplacian energy and Zagreb index of line graph, middle graph and total graph, Int. J. Contemp. Math. Sci. 5 (2010) 895-900. [73] M. C. Marino, I. Sciriha, S. K. Simić,D. V. Toˇsić,More about singular line graphs of trees, Publ. Inst. Math. (Beograd) 79(93) (2006) 1-12.

24 [74] J. McKee, C. Smyth, Salem Numbers, Pisot numbers, Mahler measure and graphs, Experi- mental Mathematics 14 (2005) 211-229.

[75] S. A. K. Merajuddin, P. Ali, S. Pirzada, Cospectral and hyper-energetic self complementary comparability graphs, J. KSIAM 11 (2007) 65-75. [76] M. Miloˇsevi´c,An example of using star complements in classifying strongly regular graphs, Filomat 22 (2008) 53-57. [77] B. Mohar, B. Tayfeh-Rezaie, Median eigenvalues of bipartite graphs, arXiv:1312.2613 (2013).

[78] A. Munemasa, Y. Sano, T. Taniguchi, Fat Hoﬀman graphs with smallest eigenvalue at least −1 − τ, Ars Math. Contemp. 7 (2014) 247-262. [79] A. Munemasa, Y. Sano, T. Taniguchi, Fat Hoﬀman graphs with smallest eigenvalue greater than −3, Discrete Appl. Math. 176 (2014),78-88.

[80] A. Munemasa, Y. Sano, T. Taniguchi, On the smallest eigenvalues of the line graphs of some trees, arXiv:1405.3475v1 (May 2014) [81] M. Petrović,T. Aleksić,S. K. Simić,Further results on the least eigenvalue of connected graphs, Linear Algebra Appl. 435 (2011) 2303-2313.

[82] M. Raˇsajski, Z. Radosavljević,B. Mihailović,Maximal reflexive cacti with four cycles: the approach via Smith graphs, Linear Algebra Appl. 435 (2011) 2530-2543. [83] Z. Radosavljević,On unicyclic reflexive graphs, Appl. Anal. Discrete Math. 1 (2007) 228-240. [84] H. S. Ramane, H. B. Walikar, S. B. Rao, B .D. Acharya, P. R. Hampiholi, S. R. Jog, I. Gutman, Spectra and energies of iterated line graphs of regular graphs, Appl. Math. Lett. 18 (2005) 679-682. [85] O. Rojo, Line graph eigenvalues and line energy of caterpillars, Linear Algebra Appl. 435 (2011) 2077-2086. [86] O. Rojo, R. D. Jiménez,Line graph of combinations of generalized Bethe trees: eigenvalues and energy, Linear Algebra Appl. 435 (2011) 2402-2419. [87] P. Rowlinson, Co-cliques and star complements in extremal strongly regular graphs, Linear Algebra Appl. 421 (2007) 157-162. [88] P. Rowlinson, On multiple eigenvalues of trees, Linear Algebra Appl. 432 (2010) 3007-3011.

[89] P. Rowlinson, On eigenvalue multiplicity and the girth of a graph, Linear Algebra Appl. 435 (2011) 2375-2381. [90] P. Rowlinson, Regular star complements in strongly regular graphs Linear Algebra Appl. 436 (2012) 1482-1488. [91] P. Rowlinson, On induced matchings as star complements in regular graphs. J. Math. Sciences 182 (2012) 159-163. [92] P. Rowlinson, On graphs with an eigenvalue of maximal multiplicity, Discrete Math. 313 (2013) 1162-1166. [93] P. Rowlinson, Star complements and connectivity in ﬁnite graphs Linear Algebra Appl. 442 (2014) 92-98. [94] P. Rowlinson , On independent star sets in ﬁnite graphs Linear Algebra Appl. 442 (2014) 82-91.

25 [95] P. Rowlinson, Eigenvalue multiplicity in cubic graphs, Linear Algebra Appl. 444 (2014) 211- 218.

[96] P. Rowlinson, Bipartite graphs with complete bipartite star complements, Linear Algebra Appl. 458 (2014) 149-160. [97] P. Rowlinson, I. Sciriha, Some properties of the Hoffman-Singleton graph, Appl. Anal. Dis- crete Math. 1(2007) 438-445. [98] P. Rowlinson, B. Tayfeh-Rezaie, Star complements in regular graphs: old and new results, Linear Algebra Appl. 432 (2010) 2230-2242. [99] I. Sciriha, Repeated eigenvalues of the line graph of a tree and of its deck, Util. Math. 71 (2006) 33-55. [100] I. Sciriha, S. K. Simić,A note on eigenspaces of some compound graphs, A tribute to Lucia Gionfriddo (Eds. M. Buratti, C. Lindner, F. Mazzocca, N. Melone), Quaderni di Matematica della Seconda Universitàdi Napoli 28 (2013) 123-154. [101] S. K. Simić,M. Andjelić,C. M. da Fonseca, D. Zivković,Onˇ the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results, Electronic J. Linear Algebra, to appear.

[102] S. K. Simić,Z. Stanić,The polynomial reconstruction is unique for the graphs whose deck- spectra are bounded from below by −2, Linear Algebra Appl. 428 (2008) 1865-1873. [103] S. K. Simić,Z. Stanić, Q-integral graphs with edge-degrees at most five, Discrete Math. 308 (2008) 4625-4634.

[104] S. K. Simić,Z. Stanić,On Q-integral (3, s)-semiregular bipartite graphs, Appl. Anal. Discrete Math. 4 (2010) 167-174. [105] S. K. Simić,D. Zivković,M.ˇ Andjelić,C. M. da Fonseca, On reflexive line graphs of trees, in preparation. [106] Z. Stanić,On graphs whose second largest eigenvalue equals 1 - the star complement technique, Linear Algebra Appl. 420 (2007) 700-710. [107] Z. Stanić,Some star complements for the second largest eigenvalue of a graph, Ars Math. Contemp. 1(2008) 126-136. [108] Z. Stanić,On nested split graphs whose second largest eigenvalue is less than 1, Linear Algebra Appl. 430 (2009) 2200-2211. [109] Z. Stanić,Some results on Q-integral graphs, Ars Combin. 90 (2009) 321-335. [110] Z. Stanić,On regular graphs and coronas whose second largest eigenvalue does not exceed 1, Linear Multilinear Algebra 58 (2010) 545-554.

[111] Z. Stanić,Graphs with small spectral gap, Electron. J. Linear Algebra 26 (2013) 417-432. [112] Z. Stanić, S. K. Simić, On graphs with unicyclic star complement for 1 as the second largest eigenvalue, Proceedings of the conference Contemporary geometry and Related Top- ics Belgrade, Serbia and Montenegro June 26 - July 2, 2005 (Eds: N. Bokan, M. Djorić, A. T. Fomenko, Z. Rakić,B. Wegner, J. Wess) pp.475-484.

[113] D. Stevanovi´c,Bipartite density of cubic graphs: the case of equality, Discrete Math. 283 (2004) 279-281.

26 [114] D. Stevanovi´c,M. Miloˇsevi´c,A spectral proof of the uniqueness of a strongly regular graph with parameters (81,20,1,6), European J. Comb. 30 (2009) 957-968.

[115] Y.-Y. Tan, Y.-Z. Fan, The vertex (edge) independence number, vertex (edge) cover number and the least eigenvalue of a graph, Linear Algebra Appl. 433 (2010) 790-795. √ [116] T. Taniguchi, On Graphs with the smallest eigenvalue at Least −1 − 2, part I, Ars Math. Contemp. 1 (2008) 81-98. √ [117] T. Taniguchi, On Graphs with the smallest eigenvalue at Least −1 − 2, part II, Ars Math. Contemp. 5 (2012) 239-254. [118] G. R. Vijayakumar, Equivalence of four description of generalized line graphs, J. Graph Theory 67 (2011) 27-33. [119] G. R. Vijayakumar, Characterization of the family of all generalized line graphs, Discuss. Math. Graph Theory 33 (2013) 637-648. [120] G. R. Vijayakumar, From ﬁnite line graphs to inﬁnite derived signed graphs, Linear Algebra Appl. 453 (2014) 84-98. [121] J.-F. Wang, Q. Huang, Spectral characterization of generalized cocktail-party graphs, J. Math. Research Appl. 32 (2012) 666-672.

[122] J.-F. Wang, Y. Shen, Q. Huang, Notes on graphs with least eigenvalue at least −2, Electron. J. Linear Algebra 23 (2012) 387-396. [123] J.-F. Wang, S. Shi, The line graphs of lollipop graphs are determined by their spectra, Linear Algebra Appl. 436 (2012) 2630-2637.

[124] J.-F. Wang, J. Yan, Comments to “The line graphs of lollipop graphs are determined by their spectra”, Linear Algebra Appl. 440 (2014) 342-344. [125] Y. Wang, Y. Z. Fan, X. X. Li, F. F. Zhang, The least eigenvalue of graphs whose complements are unicyclic, arXiv:1305.4317 (2013).

[126] Y. Wang, Y.-Z. Fan, The least eigenvalue of a graph with cut vertices, Linear Algebra Appl. 433 (2010) 19-27. [127] Y. Wang, Y.-Z. Fan, The least eigenvalue of graphs with cut edges, Graphs and Combina- torics 28 (2012) 555-561. [128] G. Yu, Y. Fan, The least eigenvalue of graphs whose complements are 2-vertex or 2-edge connected, Operations Research Trans.17 (2013) 81-88. [129] H. Yu, On the limit points of the smallest eigenvalues of regular graphs, Des. Codes Cryptogr. 65 (2012) 77-88. [130] M. Y. Yuan, Q. H. Luo, Z. K. Tang, Characterizing generalized line graphs by star complements (Chinese), J. Nat. Sci. Hunan Norm. Univ. 35 (2012) 13-16, 20. [131] H. R. Zhang, L. G. Wang, Constructing integral spectra graphs by super generalized line graph methods (Chinese), Operations Research Trans. 15 (2011) 122-128. [132] B. Zhou, A. Ili´c,On distance spectral radius and distance energy of graphs, MATCH Com- mun. Math. Comput. Chem. 64 (2010) 261-280.

[133] J. Zhou, C. Bu, Spectral characterization of line graphs of starlike trees, Linear Multilinear Algebra 61 (2013) 1041-1050.

27 [134] J. Zhou, L. Sun, H. Yao, C. Bu, On the nullity of connected graphs with least eigenvalue at least −2, Appl. Anal. Discrete Math. 7 (2013) 250-261.

(e) Other references [135] F. K. Bell, P. Rowlinson, On the multiplicities of graph eigenvalues, Bull. London Math. Soc. 35 (2003) 401-408. [136] B. Bollob´as,Combinatorics, Cambridge University Press, Cambridge, 1986.

[137] D. W. Boyd, Small Salem numbers, Duke Math. J. 44 (1977) 315-328. [138] P. J. Cameron, J.-M. Goethals, J. J. Seidel, E. E. Shult, Line graphs, root systems and elliptic geometry, J. Algebra 43 (1976) 305-327. [139] P. J. Cameron, J. H. van Lint, Designs, Graphs, Codes and their Links, Cambridge University Press, Cambridge, 1991. [140] P. D. Chawathe, G. R. Vijayakumar, A characterization of signed graphs represented by the root systen D∞, European J. Combin. 11 (1990) 523-533. [141] D. Cvetkovi´c,M. Doob, S. Simi´c,Generalized line graphs, Graph Theory, 5 (1981) 385-399.

[142] D. Cvetković,M. Petrić,A table of connected graphs on six vertices, Discrete Math. 50 (1984) 37-49. √ [143] D. Cvetković,D. Stevanović,Graphs with least eigenvalue at least − 3, Publ. Inst. Math. (Beograd), 73(87) (2003) 39-51.

[144] E. R. van Dam, W. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003) 241-272. [145] M. Doob, D. Cvetković,On spectral characterizations and embeddings of graphs, Linear Algebra Appl. 27 (1979) 17-26. [146] R. Grone, R. Merris, V. S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990) 218-238. [147] I. Gutman, I. Sciriha, On the nullity of line graphs of trees, Discrete Math. 232 (2001) 35-45. [148] W. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226-228 (1995) 593- 616. √ [149] A. J. Hoffman, −1 − 2 ?, Combinatorial Structures and Their Applications, Proc. Calgary Intern. Conf. on Combinatorial Structures and their Applications held at the Univ. of Cal- gary, June, 1969 (Eds,. R. Guy, H. Hanani, N. Sauer, J. Schönheim),Gordon and Breach Inc., New York - London - Paris, 1970, pp.173-176. [150] A. J. Hoffman, On limit points of the least eigenvalue of a graph, Ars Combin. 3 (1977) 3-14. √ [151] A. J. Hoffman, On graphs whose least eigenvalue exceeds −1 − 2, Linear Algebra Appl. 16 (1977) 153-165. [152] J. McKee, P. Rowlinson, C. Smyth, Salem numbers and Pisot numbers from stars, in: Number Theory in Progress, Proc. International Conference on Number Theory (Zakopane, Poland, 1997) (Eds. K. Gyory, H. Iwaniec, J. Urbanowicz), Walter De Gruyter (Berlin, 1999) Part 1: Diophantine Problems and Polynomials, pp. 309-319. [153] A. Neumaier, The second largest eigenvalue of a tree, Linear Algebra Appl. 46 (1982) 9-25.

28 [154] Z. Radosavljević,S. Simić,Which bicyclic graphs are reflexive?, Univ. Beograd, Publ. Elek- trotehn. Fak., Ser. Mat. 7 (1996) 90-104.

[155] B. A. Reed, ω, ∆, and χ, J. Graph Theory 27 (1998) 177-212. [156] I. Sciriha, The two classes of singular line graphs of trees, Rend. Sem. Mat. Messina, Ser. II 5 (1999) 167-180. [157] A. Torgaˇsev,A note on infinite generalized line graphs, In: Graph Theory, Proc. Fourth Yugoslav Sem. Graph Theory, Novi Sad, 15-16 April 1983, (Eds. D. Cvetković,I. Gutman, T. Pisanski, R. Toˇsić),Inst. Math. Novi Sad, 1984), pp.291-297. [158] A. Torgaˇsev,Infinite graphs with the least limiting eigenvalue greater than −2, Linear Al- gebra Appl. 82 (1986) 133-141.

[159] G. R. Vijayakumar, Signed graphs represented by D∞, European J. Combin. 8 (1987) 103-11 √ [160] R. Woo, A. Neumaier, On graphs whose smallest eigenvalue is at least −1 − 2, Linear Algebra Appl. 226-228 (1995) 577-591. [161] X. Yong, On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295 (1999) 73-80.

APPENDIX - Errata

D. Cvetkovi´c,P. Rowlinson, S. Simi´c Spectral Generalizations of Line Graphs LMS Lecture Note Series Vol. 314 Cambridge University Press ISBN 0521-83663-8 (paperback, 2004)

With acknowledgements to Bit-Shun Tam and G. R. Vijayakumar p.8 line 21: Proposition 1.1.10. Suppose that G is r-regular and U ⊆ V (G). Let U¯ = V (G) \ U. Then GU is r-regular if and only if U induces a k-regular graph and U¯ induces an h-regular graph, with |U| = 2(r − h) and |U¯| = 2(r − k). p.36 line 5: To ensure orthogonality of U, we may take Hˆ to be a tree with one petal added; this follows from Lemma 2.3.11. However the proof of Lemma 2.3.7 is incomplete because the GLG determined by Ri may have components that are trees. The assertion of Lemma 2.3.7 remains true in view of Corollary 2.3.17, which is proved independently. p.42 line 7: in the other p.87 line 17: The proof of Corollary 3.7.2 is incomplete because Lemma 3.7.1 does not necessarily apply when d vectors determine a graph with λ > −2. (A similar remark applies to Corollary 4.2 of [CvDS2].) However Corollary 3.7.2 follows from Corollary 2.3.17. p.129 line 18: [Card] p.138 line 9: star complement H0

29 p.185: the 4th and 5th graphs in Fig. 7.12b should be

@ ¨¨HH u @ ¨¨ u HH @ @ u @ u u@ u @ u u@ u @ u @ u u @ u u @ u u u u u p.188 line 5: i11 = 113 p.291 line 16: Information Processing Letters 2(1973), 108-112.