Abstract
A graph Γ is symmetric or arc-transitive if its automorphism group Aut(Γ) is transitive on the arc set of the graph, and Γ is basic if Aut(Γ) has no non-trivial normal subgroup N such that the quotient graph ΓN has the same valency as Γ. In this paper, we classify symmetric basic graphs of order 2qpn 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|(q−1), the complete graph K6 of order 6, the complete bipartite graph K5,5 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 kpn for some small integers k and n are classified.
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