Vertex and Edge Transitive, but not 1-Transitive, Graphs

Abstract

A (simple, undirected) graphGisvertex transitiveif for any two vertices ofGthere is an automorphism ofGthat maps one to the other. Similarly,Gisedge transitiveif for any two edges [a,b] and [c,d] ofGthere is an automorphism ofGsuch that {c,d} = {f(a),f(b)}. A 1-pathofGis an ordered pair (a,b) of (distinct) verticesaandbofG, such thataandbare joined by an edge.Gis 1-transitiveif for any two 1-paths (a,b) and (c,d) ofGthere is an automorphismfofGsuch thatc=f(a) andd=f(b). A graph isregular of valency dif each of its vertices is incident with exactlydof its edges.