Inverse degree and edge-connectivity

Let G be a connected graph with vertex set V ( G ) , order n = | V ( G ) | , minimum degree δ and edge-connectivity λ . Define the inverse degree of G 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 + 2 δ ( δ + 1 ) + n − 2 δ ( n − δ − 2 ) ( n − δ − 1 ) , then λ = δ . We also give an analogous result for triangle-free graphs.