Arc-transitive Cayley graphs on nonabelian simple groups with prime valency
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≤ 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≥4. Even for this problem, it was only solved for the cases when either d≤ 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≥ 11 is a prime and the vertex stabilizer is solvable.