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Star complements and exceptional graphs
Abstract
Let G be a finite graph of order n with an eigenvalue μ of multiplicity k. (Thus the μ-eigenspace of a (0,1)-adjacency matrix of G has dimension k.) A star complement for μ in G is an induced subgraph G-X of G such that |X|=k and G-X does not have μ as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, whose eigenvalues lie in [-2,∞). We establish some properties of star complements, and of eigenvectors, of exceptional graphs with least eigenvalue −2.
... To find the regular graphs with H as a star complement for µ, it clearly suffices to consider the subgraph Γ * (H, µ) of Γ(H, µ) induced by those vectors b for which b, j = −1; this is called the non-main compatibility graph in [14]. For example, Γ * (C 5 , 1) = K 5 , and the unique regular graph with C 5 as a star complement for 1 is the Petersen graph. ...
We survey results concerning star complements in finite regular graphs, and note the connection with designs and strongly regular graphs in certain cases. We include improved proofs along with new results on stars and windmills as star complements.
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.
Line graphs have the property that their least eigenvalue is greater than or equal to –2, a property shared by generalized line graphs and a finite number of so-called exceptional graphs. This book deals with all these families of graphs in the context of their spectral properties. The authors discuss the three principal techniques that have been employed, namely 'forbidden subgraphs', 'root systems' and 'star complements'. They bring together the major results in the area, including the recent construction of all the maximal exceptional graphs. Technical descriptions of these graphs are included in the appendices, while the bibliography provides over 250 references. This will be an important resource for all researchers with an interest in algebraic graph theory.
We present several results on the structure of the eigenspace of the eigenvalue -2 in generalized line graphs and in exceptional graphs; some are new and some are old but from new view points.
An exceptional graph is a connected graph with least eigenvalue greater than or equal to 2 which is not a generalized line graph. There are finitely many exceptional graphs. Maximal exceptional graphs have been recently identified. In this paper we discuss some details related to the construction of maximal exceptional graphs.
We consider some open questions about graphs with least eigenvalue -2. We investigate in particular the star complements for -2 of such graphs.
Every symmetric matrix has an invertible principal submatrix whose rank is equal to that of the whole matrix. We explore some of the implications of this result for graph spectra. Using the same symbol for a graph and its adjacency matrix, we say that a graph G is λ-invertible if G-λI is invertible, and the λ-rank of G is the rank of G-λI. A λ-basic subgraph of G is an induced λ- invertible subgraph H of the same λ-rank as G. Using λ-basic subgraphs, we prove that if λ∉{0,-1} then the set of graphs of a given λ-rank is finite; this can be extended to λ=0 and λ=-1 by excluding graphs with what we call duplicate and coduplicate vertices respectively. We give an algorithm to construct the graphs of a given λ-rank. We show how 0-basic subgraphs can be used to calculate ranks of graphs, using as an example graphs obtained by adding two vertices to a complete graph. We examine some properties of maximal reduced graphs, graphs which occur in characterising the graphs of a given rank, and construct some infinite families of such graphs.
This article is a survey of results concerning star complements in finite graphs, categorized as follows: (1) introduction, (2) construction and characterization, (3) bounds for eigenvalue multiplicities, (4) foundations of graphs, (5) exceptional graphs.
From the Publisher:
Although design theory, graph theory and coding theory, had their origins in various areas of applied mathematics, today they are found under the umbrella of discrete mathematics. Here the authors have considerably reworked and expanded their earlier successful books on designs, graphs and codes, into an invaluable textbook. They do not seek to consider each of these three topics individually, but rather to stress the many and varied connections between them. The discrete mathematics needed is developed in the text, making this book accessible to any student with a background of undergraduate algebra. Many exercises and useful hints are included throughout, and a large number of references are given.
This paper presents a variety of results on graph spectra. The number of main eigenvalues of a graph is shown to be equal to the rank of an associated matrix. We establish a condition for a graph to have exactly two main eigenvalues and then show how to evaluate them and their associated eigenvectors. It is shown that the main eigenvalues and corresponding eigenvectors of a graph determine those of its complement. We generalize to any eigenvalue a condition for 0 and -1 to be eigenvalues of a graph and its complement, respectively. Finally, we generalize to non-simple eigenvalues a result on the components of an eigenvector associated with a simple eigenvalue.
We investigate the relationship between the structure of a graph and its eigenspaces. The angles between the eigenspaces and the vectors of the standard basis of n play an important role. The key notion is that of a special basis for an eigenspace called a star basis. Star bases enable us to define a canonical bases of n associated with a graph, and to formulate an algorithm for graph isomorphism.
The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G. In this paper all graphs with λ(G) greater than −2 are characterized. Such a graph is a generalized line graph of the form L(T;1,0,…,0), L(T), L(H), where T is a tree and H is unicyclic with an odd cycle, or is one of 573 graphs that arise from the root system E8. If G is regular with λ(G)>−2, then Gis a clique or an odd circuit. These characterizations are used for embedding problems; λR(H) = sup{λ(G)z.sfnc;HinG; Gregular}. H is an odd circuit, a path, or a complete graph iff λR(H)> −2. For any other line graph H, λR(H) = −2. A similar result holds for complete multipartite graphs.
This paper summarizes the known results on graphs with smallest eigenvalue around - 2 , and completes the theory by proving a number of new results, giving comprehensive tables of the finitely many exceptions, and posing some new problems. Then the theory is applied to characterize a class of distance-regular graphs of large diameter by their intersection array.
The chapter discusses star-closed sets of lines at 60° and 90°, leading to a theorem that leaves only a restricted number of possibilities, of a specific structure. These possibilities are realized by the root systems An, Dn, E8, E7, E6 defined in terms of lines. The chapter discusses the relations to the official Root Systems, as they occur in geometry and algebra. The chapter also discusses the graphs represented by subsets of the root systems: line graphs of complete bipartite graphs for An, Hoffman's generalized line graphs for Dn, and various exceptional graphs for E8. The chapter also presents the application to Hadamard matrices.
We study the phenomenon of cospectrality in generalized line graphs and in exceptional graphs. We survey old results from today's point of view and obtain some new results partly by the use of computer. Among other things, we show that a connected generalized line graph L(H) has an excep- tional cospectral mate only if its root graph H, assuming it is itself connected, has at most 9 vertices. The paper contains a description of a table of sets of cospectral graphs with least eigenvalue at least ¡2 and at most 8 vertices together with some comments and theoretical explanations of the phenomena suggested by the table.
We now come to the central topic of the book. The first section may be viewed as a short introduction to the subject. Although we shall develop large parts of the theory of distance-regular graphs independently of Chapter 2, we shall use concepts and results about association schemes for more specialized topics such as Q-polynomial orderings (Chapter 8) and codes in graphs (Chapter 11). In §4.2 we look at various constructions that, given a distance-regular graph, produce a new one. In §4.3 we show how certain conditions on the parameters force the presence of substructures, like lines or Petersen subgraphs. In §4.4 we use the results of Chapter 3 to obtain a characterization by parameters of the two most basic families of distance-regular graphs, the Johnson and Hamming graphs. Chapter 5 contains most of the known conditions on the parameters, Chapter 6 classifies the known distance-regular graphs in various families, Chapter 7 is concerned with distance-transitive graphs, Chapter 8 discusses the consequences of the Q-polynomial property, and the remaining chapters give all examples known to us.
Let G be a connected graph with least eigenvalue –2, of multiplicity k. A star complement for –2 in G is an induced subgraph H = G – X such that |X| = k and –2 is not an eigenvalue of H. In the case that G is a generalized line graph, a characterization of such subgraphs is used to decribe the eigenspace of –2. In some instances, G itself can be characterized by a star complement. If G is not a generalized line graph, G is an exceptional graph, and in this case it is shown how a star complement can be used to construct G without recourse to root systems.
Two new graph-theoretical methods, (A) and (B), have been devised for generation of eigenvectors of weighted and unweighted
chemical graphs. Both the methods show that not only eigenvalues but also eigenvectors have full combinatorial (graph-theoretical)
content. Method (A) expresses eigenvector components in terms of Ulam’s subgraphs of the graph. For degenerate eigenvalues
this method fails, but still the expressions developed yield a method for predicting the multiplicities of degenerate eigenvalues
in the graph-spectrum. Some well-known results about complete graphs (K
n) and annulenes (C
n
), viz. (i)K
n has an eigenvalue −1 with (n−1)-fold degeneracy and (ii) C
n
cannot show more than two-fold degeneracy, can be proved very easily by employing the eigenvector expression developed in
method (A). Method (B) expresses the eigenvectors as analytic functions of the eigenvalues using the cofactor approach. This
method also fails in the case of degenerate eigenvalues but can be utilised successfully in case of accidental degeneracies
by using symmetry-adapted linear combinations. Method (B) has been applied to analyse the trend in charge-transfer absorption
maxima of the some molecular complexes and the hyperconjugative HMO parameters of the methyl group have been obtained from
this trend.
A graph is said to be exceptional if it is connected, has least eigenvalue greater than or equal to -2, and is not a generalized line graph. Such graphs are known to be representable in the exceptional root system E-8. We determine the maximal exceptional graphs by a computer search using the star complement technique, and then show how they can be found by theoretical considerations using a representation of E-8 in R-8. There are exactly 473 maximal exceptional graphs. (C) 2002 Elsevier Science (USA).
We survey the main results of the theory of graphs with least eigenvalue −2 starting from late 1950s (papers by A.J. Hoffman et al.), via important results (P.J. Cameron et al., J. Algebra 43 (1976) 305) involving root systems, to the recent approach by the star complement technique which culminated in finding and characterizing maximal exceptional graphs. Some novel results on maximal exceptional graphs are included as well. In particular, we show that all exceptional graphs, except for the cone over L(K8), can be obtained by the star complement technique starting from a unique (exceptional) star complement for the eigenvalue −2.
Suppose that the positive integer μ is the eigenvalue of largest multiplicity in an extremal strongly regular graph G. By interlacing, the independence number of G is at most 4μ2 + 4μ − 2. Star complements are used to show that if this bound is attained then either (a) μ = 1 and G is the Schläfli graph or (b) μ = 2 and G is the McLaughlin graph.
A graph is said to be exceptional if it is connected, has least eigenvalue greater than or equal to and is not a generalized line graph. Such graphs are known to be representable in the exceptional root system E 8 . The 473 maximal exceptional graphs have been found by computer using the star complement technique, and subsequently described using properties of E 8 . Here we present some information about these graphs obtained in the computer search: the exceptional star complements, some data on extendability graphs and the corresponding maximal graphs, the maximal exceptional graphs and some of their invariants. 1
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