For two vertices u and v of a graph G, the usual distance d(u, v), is the length of the shortest path between u and v. In this paper we introduced the concept of d^d-distance by considering the degrees of various vertices presented in the path, in addition to the length of the path. We study some properties with this new distance. We define the eccentricities of vertices, radius and diameter of G with respect to the d^d-distance. First we prove that the new distance is a metric on the set of vertices of G. We compare the usual, geodesic and d^d-distances of two vertices u, v of V.

Keywords: Geodesic distance, d^d-distance, d^d-Eccentricity, d^d-Radius and d^d-Diameter