Restricted edge connectivity of edge transitive graphs

Let G = (V, E) be a connected graph and S C E. S is said to be an r-restricted edge cut if G - S is disconnected and each component in G - S contains at least r vertices. Define A(r) (G) to be the minimum size of all r-restricted edge cuts and xi(r) (G) = min{omega(U): U subset of V, vertical bar U vertical bar = r and the subgraph of G induced by U is connected}, where w(U) denotes the number of edges with one end in U and the other end in V\U. A graph G with lambda((r)) (G) = xi(r)(G) (r = 1, 2,3) is called a lambda((3))-optimal graph. In this paper, we show that the only edge-transitive graphs which are not lambda((3))-optimal are the star graphs K-1,K-n-1, the cycles C-n and the cube Q(3). Based on this, we determine the expressions of N-i(G) (i = 0, 1,..., xi(3)(G) - 1) for edge transitive graph G, where N-i(G) denotes the number of edge cuts of size i in G.