Arc-transitive Cayley graphs on non-abelian simple groups with soluble vertex stabilizers and valency seven

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

“…In this paper, we shall prove the following theorem which generalizes the result in [17] to all prime valent cases, and hence gives a solution of Problem 1.2 for the case when d is a prime and the vertex-stabilizer is solvable. By Theorem 1.4, (p, 1, 1) is conceivable for each prime p ≥ 5, and by [6], (5, 4, 2) is conceivable, but not (5, 2, 2).…”

“…More recently, Pan et al [17] considered Problem 1.2 for the case when d = 7, and they proved that for a 7-valent Cayley graph Γ of a non-abelian simple group G with solvable vertex stabilizer, either Γ is normal, or Aut(Γ ) has a normal arc-transitive nonabelian simple subgroup T such that G < T and (G, T ) = (A 6 , A 7 ), (A 20 , A 21 ), (A 62 , A 63 ) or (A 83 , A 84 ), and for each of these 4 pairs (G, T ), there do exist a 7-valent G-regular T -arc-transitive graph.…”

“…By Theorem 1.4, (p, 1, 1) is conceivable for each prime p ≥ 5, and by [6], (5, 4, 2) is conceivable, but not (5, 2, 2). For the case p = 7, it was shown in [17] that (7, 1, 1), (7, 3, 1), (7,3,3) and (7,6,2) are the only conceivable triples.…”

2-Arc-transitive Cayley graphs on alternating groups

“…From the above mentioned results in [6,7,20,29,30,31] on s-arc-transitive nonnormal Cayley graphs on nonabelian simple groups of certain valencies one can observe an interesting phenomenon: most of these graphs turn out to be Cayley graphs on alternating groups. This motivates a natural problem as follows.…”

“…Du and Feng [6] proved that if Val(Γ) = 4 and s ≥ 2, then Γ is an A n+1 -arc-transitive Cayley graph on A n for n ∈ {24, 36, 72, 144, 532, 14364} with a unique exception. Similarly, Du, Feng and Zhou [7] showed that if Val(Γ) = 5 then Γ is an A n+1 -arc-transitive Cayley graph on A n where n is among 11 possible numbers, and [20] showed that if Val(Γ) = 7 and the vertex stabilizer is solvable then Γ is an A n+1 -arc-transitive Cayley graph on A n with n ∈ {7, 21, 63, 84}. Very recently, Yin, Feng, Zhou and Chen [31] proved that if Val(Γ) is a prime greater than 7 and the vertex stabilizer is solvable, then Γ is either an A n+1 -arc-transitive Cayley graph on A n or one of the three exceptions.…”