When can one see from the spectrum of a graph whether it is distance-regular or not?We give some new results for when this is the case.As a consequence we find (among others) that the following distance-regular graphs are uniquely determined by their spectrum: The collinearity graphs of the...

AMS classifications: 05E30; 51E12

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...

We characterize the distance-regular graphs with diameter three by giving an expression for the number of vertices at distance two from each given vertex, in terms of the spectrum of the graph.

A graph G has constant u = u(G) if any two vertices that are not adjacent have u common neighbours. G has constant u and u if G has constant u = u(G), and its complement G has constant u = u(G). If such a graph is regular, then it is strongly regular, otherwise precisely two vertex degrees...