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

In 2011, Fang et al. in [9] 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.