Super Edge-connectivity Of Two Classes Of Transformation Graphs

With the rapid development of communication networks, many theoretical problems have come into focus, one of which is the reliability of the network. A network is often modelled as a graph. The classical measure of the reliability is the connectivity and the edge-connectivity. For further study, many variations have been introduced, which are known as higher connectedness, such as max-λ(max-κ), super-λ(super-κ) and k-restricted edge-connectivity, etc. In another hand, to improving reliability of the network, one of methods is to constructing these higher connectedness networks.In generality, λ(G) ≤ λ(G) for any graph G where δ(G) is the minimum degree of G. A graph G is said to have maximal edge-connectivity or simply G is max — λ if λ(G) = δ(G). A graph G is said to have super edge-connectivity or simply G is super — λ if G is max — λ and every minimum edge-cut set of G isolates one vertex. In this paper, we study maximal edge-connectivity and super edge-connectivity of two classes of transformation graphs.For transformation graphs G^xyz, we contain the following main results:(1) G⁺⁺⁺ is super-A if and only if one of the following two conditions applies:(a) λ(G) ≥ 2, if G has a cut vertex x with d_G(x) = δ(G), then there are 3 or more edges between x and any component of G-x.(b) λ(G) — 1, if e=xy is a bridge, then min{d_G(x), d_G(y)} ≥2δ(G).(2) For a given graph G which has at least two edges, G^++- is super-A if and only if G (?) 2K₂ ∪ mK₁ K_1,2 ∪ mK₁, K₃ ∪ mK₁, 2K₃, G (?) K₂∪K₃, K₂∪P₃, P₄ where rn is non-negative integer.(3) For a given graph G which has no isolated vertices, G^+-+ is super-λ if and only if G has no isolated edges and G (?) K_1,n.