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 $S\setminus T$. 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 that Bush-type Hadamard matrices of order $16n^2$ give rise to strongly regular $3$-e.c. graphs, for each odd $n$ for which $4n$ is the order of a Hadamard matrix.