Connectivity of vertex and edge transitive graphs

A graph is said to be super-connected if every minimum vertex cut isolates a vertex. A graph is said to be hyper-connected if each minimum vertex cut creates exactly two components, one of which is an isolated vertex. It is proved that a connected vertex and edge transitive graph is not super-connected if and only if it is isomorphic to the lexicographic product of a cycle C-n (n greater than or equal to 6) or the line graph L(Q(3)) of the cube Q(3) by a null graph N-m. In addition, non-hyper-connected vertex and edge transitive graphs are also characterized. Precisely stated, a connected vertex and edge transitive graph G is not hyper-connected if and only if either G congruent to C-n (n greater than or equal to 6) or G congruent to L(Q(3)), or there exists a pair of vertices having the same neighbor sets and the number of vertices of G is at least k + 3, where k is the (regular) degree. (C) 2003 Elsevier Science B.V. All rights reserved.