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.