Semisymmetric Graphs of Order 2 p3 with Valency p2

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph Formula: see text of order Formula: see text, Formula: see text a prime, it is well known that Formula: see text is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group Formula: see text acting unfaithfully on at least one bipart of Formula: see text, which shows that there is only one infinite family of such graphs with valency Formula: see text. The graphs of this kind have been determined when Formula: see text acts faithfully and primitively on at least one bipart of Formula: see text, and thus there is only one remaining case for classifying such graphs of valency Formula: see text, Formula: see text acting faithfully and imprimitively on both biparts of Formula: see text, which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order Formula: see text with valency Formula: see text.