On the prime spectrum of an automorphism group of an ${\rm AT4}(p,p+2,r)$-graph

TL;DR

This paper investigates the automorphism groups of a special class of graphs called ${\rm AT4}(p,p+2,r)$-graphs, providing bounds on their prime spectra and restrictions on their structure, especially when $p$ is a prime power.

Contribution

It establishes upper bounds for the prime spectrum of automorphism groups and restrictions on their structure for ${\rm AT4}(p,p+2,r)$-graphs, advancing classification efforts.

Findings

No arc-transitive ${\rm AT4}(p,p+2,r)$-graphs exist for $p=11,17,27$

Provides bounds on prime spectra of automorphism groups

Restrictions on automorphism group structures when $p$ is a prime power

Abstract

This paper is devoted to the problem of classification of ${\rm AT4}(p,p+2,r)$-graphs. There is a unique ${\rm AT4}(p,p+2,r)$-graph with $p=2$, namely, the distance-transitive Soicher graph with intersection array $\{56, 45, 16, 1;1, 8, 45, 56\}$, whose local graphs are isomorphic to the Gewirtz graph. It is still unknown whether an ${\rm AT4}(p,p+2,r)$-graph with $p>2$ exists. The local graphs of each ${\rm AT4}(p,p+2,r)$-graph are strongly regular with parameters $((p+2)(p^2+4p+2),p(p+3),p-2,p)$. In the present paper, we find an upper bound for the prime spectrum of an automorphism group of a strongly regular graph with such parameters, and we also obtain some restrictions for the prime spectrum and the structure of an automorphism group of an ${\rm AT4}(p,p+2,r)$-graph in case when $p$ is a prime power. As a corollary, we show that there are no arc-transitive ${\rm AT4}(p,p+2,r)$-graphs with $p\in \{11,17,27\}$.