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 occur. We shall prove that a graph has constant u and u if and only if it has two distinct restricted Laplace eigenvalues. Bruck-Ryser type conditions are found. Several constructions are given and characterized. A list of feasible parameter sets for graphs with at most 40 vertices is generated.
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Paper provided by Tilburg University, Faculty of Economics and Business Administration in its series Research Memorandum with number
688.
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