A classification of semisymmetric graphs of order \(2p^3\): unfaithful case
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 3:215232, 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 28:26852715, 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 36:393405, 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).
semisymmetric graph permutation group vertex-transitive graph