Graphs with constant μ and

A graph G has constant μ − μ(G) if any two vertices that are not adjacent have μ common neighbours. G has constant μ and μ if G has constant μ = μ(G), and its complement G has constant μ = μ(G). If such a graph is regular, then it is strongly regular, otherwise precisely two vertex degrees occur. We shall prove that a connected graph has constant μ and μ if and only if it has two distinct nonzero Laplace eigenvalues. This leads to strong conditions for existence. Several constructions are given and characterized. A list of feasible parameter sets for graphs with at most 40 vertices is generated.