Finite Primitive Two-arc Transitive Graphs of Square-free Order

By analyzing the subgroup structures and the vertex-stabilizer subgroup of the graph automorphism group, the 2-arc transitive graphs of square-free order are depicted and classified, and with the method of the maximal subgroup analyzing, it is proved that such a graph is isomorphic to complete graph, double truncated Witt graph, or one of the other four coset graphs constructed.