In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency...

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,...

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. These graphs are a natural generalization of (v, k, ⋋)-graphs. In this paper we develop some theory, find many parameter conditions and give several constructions

We construct graphs that are cospectral but nonisomorphic with Kneser graphs K(n, k), when n =3k - 1, k> 2 and for infinitely many other pairs (n, k). We also prove that for 3 ≤ k ≤ n - 3 the Modulo-2 Kneser graph K2(n, k) is not determined by the spectrum

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m^4 for every positive integer m. If m is greater than 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m^4, 2m^4 + m^2, m^4 + m^2, m^4 +...

The present article is designed to be a contribution to the chapter 'Combinatorial Matrix Theory and Graphs' of the Handbook of Linear Algebra, to be published by CRC Press. The format of the handbook is to give just definitions, theorems, and examples; no proofs. In the five sections given...

For every rational number x 2 (0; 1), we construct a pair of graphs one regular and one nonregular with adjacency matrices A1 and A2, having the property that A1-xJ and A2-xJ have the same spectrum (J is the all-ones matrix). This solves a problem of Van Dam and the second author.For some values...

Let k be a natural number and let G be a graph with at least k vertices. A.E. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most e(G) (k choose 2), where e(G) is the number of edges of G. We prove this conjecture for k = 2. We also show that if G is a tree,...