Detour polynomial and wiener polynomial of Y-Tree and F-Tree

Let G(p/q) be a simple graph with p vertices and q edges. For any two vertices u and v in a connected graph G, the Detour Distance D(u,v) from u to v is defined as the length of a longest u-v path in G. The Detour Distance Polynomial or Detour Polynomial of G(p,q) is denoted by DDP(G:q) or DP(G:q) and defined as DP(G:q) = ∑q^D(U,V), the sum is taken over all unordered distinct pairs of vertices u.v in V(G). The Detour sum D(G) = DP′(G:l) where ′ denotes the first order differentiation of DP(G:q) with respect to q. Let d(u, v) denote the shortest distance between two vertices u.v ε V(G). The Wiener Polynomial of a graph G with q edges is denoted by W(G:q) and defined as W(G:q) = ∑q^{d(u, v)}, where u.v ε V(G) and the sum is taken over all unordered distinct pairs of vertices u.v in V(G). In this paper, the Detour Polynomial and Wiener Polynomial of some special trees are obtained.