The maximum distance between two subsets Ś and S of vertex set V(G) of a connected graph G is maximum distance between any two vertices u and v such that u belong to Ś and v belong to S. In this paper, we take special case of maximum distance when Ś consist of one vertex and S consist of (n – 1) vertices, n ≥ 3,. This distance is defined by: Where p is the order of a graph G. We founded Mn – polynomials, Mn – index, Hosoya polynomial and Wiener index for some special graphs such as: cog-complete, cog-star, cog-wheel, cog-path and cog-cycle.