Edge-primitive Graphs For An Almost Simple Group With Socle PSL(3,q) Of Automorphism Group

The theory of permutation groups is an important way to be used to study problems in the graph theory, and the study of symmetric graphs has been one of the main themes in algebraic graph theory. The symmetry of graphs is mainly described by the actions of the full automorphism groups of a graph on its subgraphs. For instance, the action of the full automorphism group of a graph on the point set, edge set or s-arc set to describe the symmetry of graphs. A Graph is said to be edge-transitive if its automorphism group acts transitive on the set of edges. Currently the study of edge-transitive graphs is a popular topic in algebraic graph theory. A graph is said to be edge-primitive if its automorphism group acts primitive on the set of edges. The study of edge-primitive graphs was initiated by Weiss who was a famous expert in algebraic graph theory. One of the main problems on edge-primitive graphs is to classify all edge-primitive graphs for a given automorphism group. In 2003, Guidici and Li determined all the edge-primitive graphs admitting an almost simple group with socle PSL(2,q), that is, the 2-dimensional projective linear groups. As an naturally promotion of their work, the aim of this study is to determine all edge-primitive graphs admitting an almost simple group with socle PSL(3,q). We use the information of maximal subgroups of the 3-dimensional projective linear groups, and finally a complete classification of such graphs is successfully obtained.