On the Stabilizer of the Automorphism Group of a 4-valent Vertex-transitive Graph with Odd-prime-power Order

Abstract Let X be a 4-valent connected vertex-transitive graph with odd-prime-power order p^k (kS1), and let A be the full automorphism group of X. In this paper, we prove that the stabilizer Av of a vertex v in A is a 2-group if p p 5, or a {2,3}-group if p = 5. Furthermore, if p = 5 |Av| is not divisible by 3^2. As a result, we show that any 4-valent connected vertex-transitive graph with odd-prime-power order p^k (kS1) is at most 1-arc-transitive for p p 5 and 2-arc-transitive for p = 5.