On the $$\mathrm{PSU}(4,2)$$-Invariant Vertex-Transitive Strongly Regular (216, 40, 4, 8) Graph

Abstract

In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a strongly regular graph $$\Gamma$$ with parameters (216, 40, 4, 8) and proved that it is the unique $$\mathrm{PSU}(4,2)$$-invariant vertex-transitive graph on 216 vertices. In this paper, using the geometry of the Hermitian surface of $$\mathrm{PG}(3,4)$$, we provide a computer-free proof of the existence of the graph $$\Gamma$$. The maximal cliques of $$\Gamma$$ are also determined.