Is Taylor's graph geometric?

Abstract

Taylor's graph is a strongly regular graph that is the unique descendant of a certain regular two-graph on q 3 + 1 points, where q is an odd prime power. It has the parameters of the point graph of a putative partial geometry PG(q − 1, 1 2 (q 2 − 1), 1 2 (q − 1)) and so is pseudo-geometric. Here we investigate the question as to whether or not Taylor's graph is geometric and discover that it is when q = 3 but not in the cases q = 5, 7.