Hosoya Polynomials and Wiener Indices of Distances in Graphs

In this work, we deal with three types of distances, namely ordinary distance, the minimum distance (n-distance), and the width distance (w-distance). The ordinary distance between two distinct vertices u and v in a connected graph G is defined as the minimum of the lengths of all u-v paths in G, and usually denoted by dG(u, v), or d(u, v).The minimum distance in a connected graph G between a singleton vertex v belong to V and (n-1)-subset S of V, n 2, denoted by dn(u, v) and termed n-distance, is the minimum of the distances from v to the vertices in S.The container between two distinct vertices u and v in a connected graph G is defined as a set of vertex-disjoint u-v paths, and is denoted by C(u, v). The container width w = w(C(u, v)), is the number of paths in the container, i.e., w(C(u, v)) = -C(u.v)-. The length of a container l = l(C(u, v)) is the length of a longest path in C(u, v).For every fixed positive integer w, the width distance (w-distance) between u and v is defined as: dn* (u, v-G)= min l(C(u, v)), where the minimum is taken over all containers C(u, v) of width w. Assume that the vertices u and v are distinct when w 2.