Willem H. Haemers

Interlacing eigenvalues and graphs

Abstract We give several old and some new applications of eigenvalue interlacing to matrices associated to graphs. Bounds are obtained for characteristic numbers of graphs, such as the size of a maximal (co)clique, the chromatic number, the diameter, and the bandwidth, in terms of the eigenvalues of the standard adjacency matrix or the Laplacian matrix. We also deal with inequalities and regularity results concerning the structure of graphs and block designs.

Enumeration of cospectral graphs

We have enumerated all graphs on at most 11 vertices and determined their spectra with respect to various matrices, such as the adjacency matrix and the Laplacian matrix. We have also counted the numbers for which there is at least one other graph with the same spectrum (a cospectral mate). In addition we consider a construction for pairs of cospectral graphs due to Godsil and McKay, which we call GM switching. It turns out that for the enumerated cases a large part of all cospectral graphs comes from GM switching, and that the fraction of graphs on n vertices with a cospectral mate starts to decrease at some value of n < 11 (depending on the matrix). Since the fraction of cospectral graphs on n vertices constructible by GM switching tends to 0 if n → ∞, the present data give some indication that possibly almost no graph has a cospectral mate. We also derive asymptotic lower bounds for the number of graphs with a cospectral mate from GM switching.

On Some Problems of Lov&#225;sz Concerning the Shannon Capacity of a Graph

The answers to several problems of Lov\hat{a}sz concerning the Shannon capacity of a graph are shown to be negative.

Strongly Regular Graphs

A graph (simple, undirected, and loopless) of order v is called strongly regular with parameters v, k,λ,μ whenever it is not complete or edgeless.

The Gewirtz Graph

We prove that there is a unique graph (on 56 vertices) with spectrum 101235(-4)20 and examine its structure. It turns out that, e.g., the Coxeter graph (on 28 vertices) and the Sylvester graph (on 36 vertices) are induced subgraphs. We give descriptions of this graph.

Strongly Regular Graphs with Maximal Energy

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Koolen and Moulton have proved that the energy of a graph on n vertices is at most n(1 + √n)/2, and that equality holds if and only if the graph is strongly regular with parameters (n, (n+√n)/2, (n+2√n)/4, (n+2√n)/4). Such graphs are equivalent to a certain type of Hadamard matrices. Here we survey constructions of these Hadamard matrices and the related strongly regular graphs.

The complement of the path is determined by its spectrum

Abstract It is proved that a graph whose (0,1) -adjacency matrix has the spectrum of P n , the complement of the path on n vertices, must be P n .

Binary Codes of Strongly Regular Graphs

For strongly regular graphs ith adjacency matrix A, we look at the binary codes generated by A and A + I. We determine these codes for some families of graphs, e pay attention to the relation beteen the codes of switching equivalent graphs and, ith the exception of two parameter sets, we generate by computer the codes of all knon strongly regular graphs on fewer than 45 vertices.

Spreads in Strongly Regular Graphs

A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsartes bound (also called Huffmans bound). Such spreads give rise to colorings meeting Hoffmans lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences lead to conditions for existence. Most examples come from spreads and fans in (partial) geometries. We give other examples, including a spread in the McLaughlin graph. For strongly regular graphs related to regular two-graphs, spreads give lower bounds for the number of non-isomorphic strongly regular graphs in the switching class of the regular two-graph.

Deza graphs: A generalization of strongly regular graph

We consider the following generalization of strongly regular graphs. A graph G is a Deza graph if it is regular and the number of common neighbors of two distinct vertices takes on one of two values (not necessarily depending on the adjacency of the two vertices). We introduce several ways to construct Deza graphs, and develop some basic theory. We also list all diameter two Deza graphs which are not strongly regular and have at most 13 vertices.