Inverse degree and super edge-connectivity

摘要

Let G be a connected graph of order n, minimum degree δ(G) and edge connectivity λ(G). The graph G is called maximally edge-connected if λ(G) = δ(G), and super edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree. Define the inverse degree of G with no isolated vertices as R(G) = ∑_(v∈V(G))1/d(v). Where d(v) denotes the degree of the vertex v. We show that if R(G) < 2 + (n - 28δ)/(n - δ)(n - δ - 1), then G is super edge-connected. We also give an analogous result for triangle-free graphs.