Semisymmetric prime-valent graphs of order $$2p^3$$

Abstract

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime and $$\Gamma $$ a semisymmetric prime-valent graph of order $$2p^3$$ . Then, $$\Gamma $$ is bipartite. Denote by $$\mathrm{Aut}(\Gamma )$$ the full automorphism group of $$\Gamma $$ . In Du and Wang (J Algebraic Combin 41:275–302, 2015), Wang and Du (Eur J Combin 36:393–405, 2014) and Wang et al. (Ars Math Contemp 7:40–53, 2014), the first author and Du proved that there is no prime-valent graph when $$\mathrm{Aut}(\Gamma )$$ acts unfaithfully on at least one bipart of $$\Gamma $$ . The first author in Wang (Ars Combin 133:3–15, 2017) gave a complete classification when $$\mathrm{Aut}(\Gamma )$$ acts faithfully and primitively on at least one bipart of $$\Gamma $$ , and as a result there is only one such graph, that is, cubic Gray graph. Due to these efforts, there is only one remaining case for classifying such graphs: $$\mathrm{Aut}(\Gamma )$$ acts faithfully and imprimitively on both biparts of $$\Gamma $$ , which is dealt with in this paper. We prove that there are two infinite families of such graphs. Thus, combining these results, we can get the complete classification of semisymmetric prime-valent graphs of order $$2p^3$$ .