Arc-transitive Cayley graphs on nonabelian simple groups with prime valency

TL;DR

This paper advances the classification of non-normal locally primitive Cayley graphs on nonabelian simple groups with prime valency by solving the problem for cases where the valency is at least 11 and the vertex stabilizer is solvable.

Contribution

It provides a complete solution for classifying such Cayley graphs when the prime valency is at least 11 and the vertex stabilizer is solvable, addressing a longstanding open problem.

Findings

Solved the classification problem for prime valency d ≥ 11

Extended understanding of Cayley graphs on nonabelian simple groups

Progressed towards a comprehensive classification for various valencies

Abstract

In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency $d$, where either $d\leq 20$ or $d$ is a prime number. The only case for which the complete solution of this problem is known is of $d=3$. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valency $d\geq4$. Even for this problem, it was only solved for the cases when either $d\leq 5$ or $d=7$ and the vertex stabilizer is solvable. In this paper, we make crucial progress towards the above problems by completely solving the second problem for the case when $d\geq 11$ is a prime and the vertex stabilizer is solvable.