Let G = (V; E) be a graph and X ⊆ V . The differential of X is defined as @(X) = jB(X)j − jXj where B(X) is a set of all vertices in V − X which has adjacent vertex in the set X. Also, the differential of a graph G written @(G), is equal to maxf@(X) : X ⊆ V g. In this paper, we initiate the study of k-distance differential of graphs which is a generalization of differential of graphs. In addition, we show that for any connected graph G of order n ≥ k + 2, the number (22kk−+3 1)n is a sharp lower bound for k-distance differential of G. We also obtain upper bounds for k-distance differential of graphs. Finally, we characterize the graphs whose k-distance differential belongs to fn − 2; n − 3; 1g
