On optimally-λ(3)λ(3) transitive graphs

Let X=(V,E)X=(V,E) be a connected graph and S⊆E.S is said to be an mm-restricted edge cut if X-SX-S is disconnected and each component of X-SX-S contains at least m   vertices. Let λ(m)(X)λ(m)(X) be the minimum size of all m  -restricted edge cuts, and ξm(X)=min{ω(U): U⊆V,|U|=mU⊆V,|U|=m and X[U]X[U] is connected}, where ω(U)ω(U) is the number of edges with one end in U   and the other end in V⧹UV⧹U, X[U]X[U] is the subgraph of X induced by U. A graph X   is said to be optimally-λ(3)λ(3) if λ(i)(X)=ξi(X)(i=1,2,3). In this paper, optimally-λ(3)λ(3) vertex-transitive graphs are studied. In particular, generating sets of optimally-λ(3)λ(3) minimal Cayley graphs are completely characterized.