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

Abstract

A (simple, undirected) graph G is vertex transitive if for any two vertices of G there is an automorphism of G that maps one to the other. Similarly, G is edge transitive if for any two edges [ a , b ] and [ c , d ] of G there is an automorphism of G such that { c , d } = { f ( a ), f ( b )}. A 1- path of G is an ordered pair ( a , b ) of (distinct) vertices a and b of G , such that a and b are joined by an edge. G is 1- transitive if for any two 1-paths ( a , b ) and ( c , d ) of G there is an automorphism f of G such that c = f ( a ) and d = f ( b ). A graph is regular of valency d if each of its vertices is incident with exactly d of its edges.