The energy of a graph T is the sum of the absolute values of the eigenvalues of the adjacency matrix of T. Seidel switching is an operation on the edge set of T. In some special cases Seidel switching does not change the spectrum, and therefore the energy. Here we investigate when Seidel...

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 +...

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,...

We give sufficient conditions for existence of a perfect matching in a graph in terms of the eigenvalues of the Laplacian matrix. We also show that a distance-regular graph of degree k is k-edge-connected

Suppose one looks for a square integral matrixN, for which NN has a prescribed form.Then the Hasse-Minkowski invariants and the determinant of NN lead to necessary conditions for existence. The Bruck-Ryser-Chowla theorem gives a famous example of such conditions in case N is the incidence matrix...

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if...

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...

Consider a graph Ґ on n vertices with adjacency matrix A and degree sequence (d1, . . . , dn). A universal adjacency matrix of Ґ is any matrix in Span {A,D, I, J} with a nonzero coefficient for A, where D = diag (d1, . . . , dn) and I and J are the n × n identity and all-ones matrix,...