A classification result about basic 2-arc-transitive graphs (Q6589149)

Let \(\Gamma=(V, E)\) be a graph with vertex set \(V\) and edge set \(E\). An arc of \(\Gamma\) is an ordered pair of adjacent vertices, and a 2-arc is a triple \((\alpha, \beta, \gamma)\) of vertices with \(\{\alpha,\beta\}, \{\beta,\gamma\}\in E\) and \(\alpha\neq \gamma\). Denote by \(\mathrm{Aut}(\Gamma)\) the full automorphism group of the graph \(\Gamma\), and call every subgroup of \(\mathrm{Aut}(\Gamma)\) an (automorphism) group of \(\Gamma\). A group \(G\) of \(\Gamma\) is said to be 2-arc-transitive if \(G\) acts transitively on the \(2\)-arcs of \(\Gamma\). A connected graph \(\Gamma =(V, E)\) with at least \(3\) vertices is called a basic \(2\)-arc-transitive graph if it has a \(2\)-arc-transitive group \(G\) such that every minimal normal subgroup of \(G\) has at most two orbits on the vertex set \(V\). \N\N\textit{C. E. Praeger} observed in [J. Lond. Math. Soc., II. Ser. 47, No. 2, 227--239 (1992; Zbl 0738.05046)] and [ Australas. J. Comb. 7, 21--36 (1993; Zbl 0776.05050)] that every connected \(2\)-arc-transitive graph is a cover of some basic \(2\)-arc-transitive graph, and proposed the following problem: Classify all finite basic \(2\)-arc-transitive graphs. In the present paper, the authors give a classification for basic \(2\)-arc-transitive non-bipartite graphs of order \(r^as^b\) and basic \(2\)-arc-transitive bipartite graphs of order \(2r^as^b\), where \(r\) and \(s\) are distinct primes.