A classification of semisymmetric graphs of order $$2p^3$$ 2 p 3 : unfaithful case

Abstract

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive, but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two biparts of equal size. It was proved by Folkman in (J Comb Theory Ser B 3:215–232, 1967) that there exist no semisymmetric graphs of order $$2p$$ and $$2p^2$$ , where $$p$$ is a prime. For any distinct primes $$p$$ and $$q$$ , the classification of semisymmetric graphs of order $$2pq$$ was given by Du and Xu in (Comm Algebra 28:2685–2715, 2000). Naturally, one of our long-term goals is to determine all the semisymmetric graphs of order $$2p^3$$ , for any prime $$p$$ . All these graphs $$\Gamma $$ are divided into two subclasses: (I) the automorphism group $$\hbox {Aut }(\Gamma )$$ acts unfaithfully on at least one bipart; and (II) $$\hbox {Aut }(\Gamma )$$ acts faithfully on both biparts. In Wang and Du (Eur J Comb 36:393–405, 2014), a group theoretical characterization for Subclass (I) was given by the authors. Based on this characterization, this paper gives a complete classification for Subclass (I).