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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, fol- lowing 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 f1; : : : ; ng, 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 Schl¨afligraph, 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.
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