Inverse degree and edge-connectivity

Let GG be a connected graph with vertex set V(G)V(G), order n=|V(G)|n=|V(G)|, minimum degree δδ and edge-connectivity λλ. Define the inverse degree   of GG as R(G)=∑v∈V(G)1d(v), where d(v)d(v) denotes the degree of the vertex vv. We show that if R(G)<2+2δ(δ+1)+n−2δ(n−δ−2)(n−δ−1), then λ=δλ=δ. We also give an analogous result for triangle-free graphs.