On basic graphs of symmetric graphs of valency five

A graph \G is {\em symmetric} or {\em arc-transitive} if its automorphism group \Aut(\G) is transitive on the arc set of the graph, and \G is {\em basic} if \Aut(\G) has no non-trivial normal subgroup $N$ such that the quotient graph \G_N has the same valency with \G. In this paper, we classify symmetric basic graphs of order $2qp^n$ and valency 5, where $q<p$ are two primes and $n$ is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order $2q$ with 5\di (q-1), the complete graph $K_6$ of order $6$, the complete bipartite graph $K_{5,5}$ of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order $kp^n$ for some small integers $k$ and $n$ are classified