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Restricted detour distance between two vertices and of a graph is the length of a longest u- v path for the induced condition (V (P))=P. The restricted detour polynomial depends on restricted detour distance and is denoted by D*(G,x) and defined by D*(G,x) = ∑{u,v}xD*(u,v) where the summation is taken over all unordered pairs (u,v) of distinct vertices of u and v of V(G). In this paper, we investigate the restricted detour polynomial of the straight chain of wheel graphs φ(WkB) and compute the restricted detour index of φ(WkB).

The restricted detour distance D*(u,v) between two vertices u and v of a connected graph G is the length of a longest u - v path P in G such that = P. The main goal of this paper is to obtain the restricted detour polynomial of the theta graph. Moreover, the restricted detour index of the theta graph will also be obtained.

In this paper we will introduce a new graph distance based polynomial; Detour Hosoya polynomials of graphs . The Detour Hosoya polynomials for some special graphs such as paths and cycles are obtained. Moreover the Detour Hosoya polynomials , and are obtained.

The Hosoya polynomials of Steiner n-distance of the sequential join of graphs and are obtained and the Hosoya polynomials of Steiner 3-distance of the sequential join of m graphs are also obtained.

The Hosoya polynomials of Steiner distance of complete m-partite graphs and Straight hexagonal chains are obtained in this paper. The Steiner n-diameter and Wiener index of Steiner n-distance of and are also obtained.

ABSTRACT The Hosoya polynomials of Steiner 3-distance of hypercube graphs , and of the square of a path, , are obtained in this paper. The Steiner n-diameters of and are also obtained.