Locally Transitive Graphs Admitting a Group with Cyclic Sylow Subgroups

All graphs are finite simple undirected and of no isolated vertices in this paper.Using the theory of coset graphs and permutation groups,it is completed that a classification of locally transitive graphs admitting a non-Abelian group with cyclic Sylow subgroups.They are either the union of the family of arc-transitive graphs,or the union of the family of bipartite edge-transitive graphs.