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. in 2011 and Pan et al. in 2013 determined all pentavalent symmetric graphs of order 4 p q . 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 such graph Γ , either the full automorphism group A u t Γ is isomorphic to P S L (2, p ) , P G L (2, p ) , P S L (2, p ) × ℤ 2 or P G L (2, p ) × ℤ 2 , or Γ is isomorphic to one of 9 graphs.