Pentavalent symmetric graphs of order four times an odd square-free integer

Abstract

A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of order $4pq$. In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each of such graphs $\Gamma$, either the full automorphism group $\mathsf{Aut}\Gamma$ is isomorphic to $\mathsf{PSL}(2,p)$, $\mathsf{PGL}(2,p)$, $\mathsf{PSL}(2,p)\times {\mathbb{Z}}_2$ or $\mathsf{PGL}(2,p)\times {\mathbb{Z}}_2$, or $\Gamma$ is isomorphic to one of 8 graphs.

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