Semisymmetric Cubic Graphs Constructed from Bi-Cayley Graphs of \(A_n\)

Abstract

For a group \(T\) and a subset \(S\) of \(T\), the bi-Cayley graph \(\text{BCay}(T, S)\) of \(T\) with respect to \(S\) is the bipartite graph with vertex set \(T \times \{0, 1\}\) and edge set \(\{\{(g, 0), (ag, 1)\} | g \in T, s \in S\}\). In this paper, we investigate cubic bi-Cayley graphs of finite nonabelian simple groups. We give several sufficient or necessary conditions for a bi-Cayley graph to be semisymmetric, and construct several infinite families of cubic semisymmetric graphs.