Hadamard Matrices and Strongly Regular Graphs with the 3-e.c. Adjacency Property

A graph is 3-e.c. if for every 3-element subset S of the vertices, and for every subset T of S, there is a vertex not in S which is joined to every vertex in T and to no vertex in S \ T. Although almost all graphs are 3-e.c., the only known examples of strongly regular 3-e.c. graphs are Paley graphs with at least 29 vertices. We construct a new infinite family of 3-e.c. graphs, based on certain Hadamard matrices, that are strongly regular but not Paley graphs. Specifically, we show ∗The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) the electronic journal of combinatorics 8 (2001), #R1 1 that Bush-type Hadamard matrices of order 16n2 give rise to strongly regular 3-e.c. graphs, for each odd n for which 4n is the order of a Hadamard matrix.