Four Times Prime Power Order Of Five Degrees Symmetrical Map

Let Γ be a graph, and let AutΓ denote its full automorphism group. If AutΓ is transitive on the arc-set of Γ, then Γ is called a symmetric graph. A symmetric graph is called a basic graph if it admits an arc-transitive automorphism group which is quasiprimitive or biquasiprimitive on its vertex set. Let VΓ,EΓ, AΓ denote the vertex set, edge set, arc set of Γ, respectively. For α∈VΓ, X≤AutΓ, let Γ(a) be the set of vertices which adjacent to α, the stabilizer Xa is primitive on Γ(a), then Γ is called X-locally primitive.Characterizing arc-transitive graphs with small valency and restricted orders has received considerable attention on the literature with numerous references. For instance, see [4] for cubic arc-transitive graphs of order on up to768;[6,7] for cubic arc-transitive graphs of order2p2with p a prime;[25,26] for cubic arc-transitive graphs of order14p and16p with p a prime. Characterizing pentavalent symmetric graphs have received much attention on the literature, see, for example, pentavalent symmetric graphs of order2pq is classified in [17]; and pentavalent symmetric graphs of order8p is classified in [18]. For more results, see references [14-20].In this paper, we classify basic pentavalent symmetric graphs of order4pn with p a prime and n>1. It is shown that such graphs are normal covers of the four specific graphs K6,6-6K2,I12,Q4d or (?)3636, with orders6,12,16or36, respectively. In particuliar, there is no example for each p>5. Moreover, by using the above result, all arc-transitive pentavalent graphs of order four times a prime square and four times a prime cube are determined, consist of exactly four graphs.