Abstract :
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 Cn (n⩾6) or the line graph L(Q3) of the cube Q3 by a null graph Nm. 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≅Cn (n⩾6) or G≅L(Q3), 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.
