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)} frac(1, d (v)), where d (v) denotes the degree of the vertex v. We show that if R (G) < 2 + frac(2, δ (δ + 1)) + frac(n - 2 δ, (n - δ - 2) (n - δ - 1)), then λ = δ. We also give an analogous result for triangle-free graphs.