# On the PSU(4, 2)-invariant vertex-transitive strongly regular (216, 40, 4, 8) graph

## TL;DR

This paper provides a computer-free proof of the existence of a unique PSU(4,2)-invariant vertex-transitive strongly regular graph with parameters (216, 40, 4, 8), using Hermitian surface geometry.

## Contribution

It offers a novel geometric proof of the graph's existence and characterizes its maximal cliques, advancing understanding of this specific strongly regular graph.

## Key findings

- Existence of the graph is proven without computer assistance.
- The graph's maximal cliques are explicitly determined.
- The graph's uniqueness and symmetry properties are confirmed.

## 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 PSU(4,2)-invariant vertex-transitive graph on 216 vertices. In this paper, using the geometry of the Hermitian surface of PG(3, 4), we provide a computer-free proof of the existence of the graph $\Gamma$. The maximal cliques of $\Gamma$ are also determined.