For almost all graphs the answer to the question in the title is still unknown. Here we survey the cases for which the answer is known. Not only the adjacency matrix, but also other types ofâ€¦ (More)

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 maximalâ€¦ (More)

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 theâ€¦ (More)

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 determinedâ€¦ (More)

The values t = 1, 3, 5, 6, 9 satisfy the standard necessary conditions for existence of a generalised quadrangle of order (3, t). This gives the following possible parameter sets for strongly regularâ€¦ (More)

Brouwer, A.E, W.H. Haemers, Structure and uniqueness of the (81,20, 1.6) strongly regular graph, Discrete Mathematics 106/107 (1992) 77-82. We prove that there is a unique graph (on 81 vertices) withâ€¦ (More)

We show that if Î¼j is the j-th largest Laplacian eigenvalue, and dj is the j-th largest degree (1 â‰¤ j â‰¤ n) of a connected graph Î“ on n vertices, then Î¼j â‰¥ dj âˆ’ j + 2 (1 â‰¤ j â‰¤ nâˆ’ 1). This settles aâ€¦ (More)

We investigate when a complete graph Kn with some edges deleted is determined by its adjacency spectrum. It is shown to be the case if the deleted edges form a matching, a complete graph Km providedâ€¦ (More)