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...

2010 Mathematics Subject Classification: 05E30, 05C50;

AMS classifications; 05C50; 05E30;

AMS classifications: 05E30; 05B20

AMS classifications: 05C50; 05E99;

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...

We study the connected regular graphs with four distinct eigenvalues. Properties and feasibility conditions of the eigenvalues are found. Several examples, constructions and characterizations are given, as well as some uniqueness and nonexistence results.