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 \(2p^3\).

如果简单的无向图是规则的且具有边沿传递性但不具有顶点传递性，则称其为半对称的。令p为素数，\（\ Gamma \）为\（2p ^ 3 \）的半对称素数图。然后，\（\ Gamma \）是二分的。表示由\（\ mathrm {AUT}（\伽玛）\）的全自同构组的\（\伽玛\） 。在Du and Wang（J Algebraic Combin 41：275-302，2015），Wang and Du（Eur J Combin 36：393-405，2014）和Wang等人的著作中。（Ars Math Contemp 7：40–53，2014），第一作者和Du证明当\（\ mathrm {Aut}（\ Gamma）\）不忠实地作用于\的至少一个二分之一时，没有素数图（\ Gamma \）。Wang的第一作者（2017年Ars Combin 133：3-15）对\（\ mathrm {Aut}（\ Gamma）\）忠实且原始地作用于\（\ Gamma \）的至少两个部分时，给出了完整的分类。，因此只有一个这样的图，即立方灰色图。由于这些努力，仅剩下一种情况可以对这些图进行分类：\（\ mathrm {Aut}（\ Gamma）\）对\（\ Gamma \）的两个等分部分都忠实地，不加约束地起作用，这在本章中将进行处理。纸。我们证明了这种图有两个无限的族。因此，结合这些结果，我们可以得到\（2p ^ 3 \）阶半对称素数图的完整分类。