On the Wiener Polynomials of Some Trees

69 On the Wiener Polynomials of Some Trees Ali A. Ali Ahmed M. Ali aliazizali1933@yahoo.com ahmedgraph@uomosul.edu.iq College of Computer Sciences and Mathematics University of Mosul, Iraq Received on: 02/05/2006 Accepted on: 16/08/2006 ABSTRACT The Wiener index is a graphical invariant which has found many applications in chemistry. The Wiener Polynomial of a connected graph G is the generating function of the sequence (C(G,k)) whose derivative at x=1 is the Wiener index W(G) of G, in which C(G,k) is the number of pairs of vertices distance k apart. The Wiener Polynomials of star-like trees and other special trees are found in this paper; and hence a formula of the Wiener index for each such trees is obtained .


Introduction
Let G be a finite connected graph of vertex set V. The distance between vertices u and v in G is the length of a shortest u-v path. Let ) v , u ( d G , or simply d (u,v), denote the distance between vertices u and v. The eccentricity e(v) of a vertex v is the greatest possible distance from v to all other vertices of G, that is The diameter of G, denoted by diamG or  , is defined as The radius of G, denoted by radG or r, is defined as The Wiener index of G is defined as where the sum is over all unordered pairs {u,v} of distinct vertices in G. It is clear that The average distance of G is defined as where p is the order of G . It is know [2] that . If x is a parameter, then the Wiener polynomial of G is where the sum is taken over all unordered pairs {u,v} of vertices in G . Let d(G,k) denote the number of all distinguishable unordered pairs of vertices that are of distance k apart. Then It is clear that the Wiener index W(G) is equal to the value of the derivative of W(G;x), with respect to x , at x=1 Deriving a formula for the Wiener polynomial of some type of graphs requires that the graphs must have a particular degree of uniformity. Therefore several authors had obtained Wiener polynomials for special graphs and compound graphs (see [4] and [5]).The trees are considered as the most important and useful kind of graphs. Therefore many papers have been written about the average distance of particular graphs including trees. Since Wiener polynomials provide us with more information about distance, it is useful to find Wiener polynomials of some type of trees. In 1996, Sagan, Yeh and Zhang [6] obtained and studied Wiener polynomial for trees called "dendrimer" which are used in chemistry. In 2002, D. Bonchev and D. J. Klein [1] obtained the Wiener index of thorn rods and thorn stars that are used in theoretical chemistry. Therefore it is suitable to find Wiener polynomials for some other kinds of trees, as we have done in this paper.

The Wiener Polynomial of a Star-like
If m=1 , then ) 1 ( T is a path from the vertex c to the vertex 1 u , and of the degree 1 1  + , and using the formula of the polynomial of a path [4], we get which is the same result obtained from (2.1) when m=1. If m=2 , we use Theorem 1 of Gutman [4] , and we get ) where F is a path from the vertex c to the vertex 2 u of order 2 1  + , and F T T which is the result obtained from (2.1) when we put m=2. If m=3 we a gain use Theorem 1 of Gutman [4], and we get where Q is a path from the vertex c to the vertex 3 u of order 3 1  + , and which is the result obtained from (2.1) when we put m=3. Now assume that the formula (2.1) be true for the tree We shall prove that it's true for the tree Therefore , substituting for ,

Proof:
Differentiating (2.1) with respect to x and replacing x=1, we get , m ( F  is given by

Proof:
Using Theorem 2.1 and putting  =  j for all j=1,2,…,m, we get and its average distance is

Proof:
The , and will suppose that      .
To simplify the symbol we'll denote it by M in the following results.

Theorem 3.1: The Wiener Polynomial of a tree M is given by
We partition the set of the vertices of the tree M into three subsets It's clear that each of the induced sub graphs To explain the proof we divide it into two parts. Hence, the total number of unordered pairs of vertices that are distance k apart, when , the number of the pairs is zero. Therefore Notice that its not true in this case to write the number Second: When , the unordered pairs of vertices that are distance k apart are of the following five kinds: (1) One vertex in the set 1 V and the other in the set 2 V ; the number of the these pairs is Hence by taking the numbers of pairs in the cases (1),(2),(3),(4) and (5), we get the formula mentioned in the statement of the theorem when     k . Thus the proof is completed .

Remark
The Wiener Polynomial of the tree  (1) of Gutman [4], as it is given in the next theorem.
Notice that Theorem 3.2 is more general than Theorem 3.1 , because it doesn't require the condition      ; but it's possible to find formulas for d(M,k) when , as in Theorem 3.1. But Theorem 3.1 is more useful than Theorem 3.2 when we want to find the coefficient of a particular power of x.

Proof:
To get the Wiener index of