Lower bounds on the differential of a graph

Let G=(V,E) be a graph of order n and let B(D) be the set of vertices in V∖D that have a neighbor in a set D. The differential of a set D is defined as ∂(D)=|B(D)|−|D| and the differential of a graph to equal the maximum value of ∂(D) for any subset D of V. In this paper, we obtain several tight lower bounds for the differential of a graph.