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

## TL;DR

This paper classifies all connected pentavalent symmetric graphs of order four times an odd square-free integer, identifying their automorphism groups or specific isomorphism types.

## Contribution

It generalizes previous classifications by determining all such graphs for a broader class of orders, including explicit automorphism group structures.

## Key findings

- Automorphism groups are isomorphic to PSL(2,p), PGL(2,p), or their products with Z_2.
- Identifies 8 specific graphs with unique structures.
- Provides a complete classification for the specified order class.

## 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 $\it\Gamma$, either the full automorphism group ${\sf Aut}\it\Gamma$ is isomorphic to ${\sf PSL}(2,p)$, ${\sf PGL}(2,p)$, ${\sf PSL}(2,p){\times}\mathbb{Z}_2$ or ${\sf PGL}(2,p){\times}\mathbb{Z}_2$, or $\it\Gamma$ is isomorphic to one of 8 graphs.