\magnification=1200
\hsize=4in
\nopagenumbers
\noindent
%
%
{\bf Anthony Bonato, W. H. Holzmann, Hadi Kharaghani}
%
%
\medskip
\noindent
%
%
{\bf Hadamard Matrices and Strongly Regular Graphs
with the $3$-e.c. Adjacency Property}
%
%
\vskip 5mm
\noindent
%
%
%
%
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.


\bye