Reciprocal Complementary Wiener Numbers of Non-Caterpillars

The reciprocal complementary Wiener number of a connected graph G is defined as ( ) { } ( ) ( ) | ∑ u v V G RCW G d d u v G ⊆ = + − , 1 1 , where ( ) V G is the vertex set. ( ) | d u v G , is the distance between vertices u and v, and d is the diameter of G. A tree is known as a caterpillar if the removal of all pendant vertices makes it as a path. Otherwise, it is called a non-caterpillar. Among all n-vertex non-caterpillars with given diameter d, we obtain the unique tree with minimum reciprocal complementary Wiener number, where d n ≤ ≤ − 4 3 . We also determine the n-vertex non-caterpillars with the smallest, the second smallest and the third smallest reciprocal complementary Wiener numbers.

where ( ) V G is the vertex set.( ) is the distance between vertices u and v, and d is the diameter of G.A tree is known as a caterpillar if the removal of all pendant vertices makes it as a path.Otherwise, it is called a non-caterpillar.Among all n-vertex non-caterpillars with given diameter d, we obtain the unique tree with minimum reciprocal complementary Wiener number, where

Introduction
The Wiener number was one of the oldest topological indices, which was introduced by Harry Wiener in 1947.About the recent reviews on matrices and topological indices related to Wiener number, refer to [1]- [4].The RCW number is one of the hotest additions in the family of such descriptors.The notion of RCW number was first put forward by Ivanciuc and its applications were discussed in [5]- [8].
Let G be a simple connected graph with vertex set ( ) V G .For two vertices ( ) ( ) , | d u v G denote the distance between u and v in G.Then, the RCW number of G is defined by where d is the diameter and the summation goes over all unordered pairs of distinct vertices of G. Some properties of the RCW number have been obtained in [9] [10].
A tree is called a caterpillar if the removal of all pendant vertices makes it as a path.Otherwise, it is called a non-caterpillar.
For integers n and d satisfying 4 be the tree obtained from the path by attaching the path 2 P and 3 .
In this paper, we show that among all n-vertex non-caterpillars with given diameter d, is the unique tree with minimum RCW number where 4 3 d n ≤ ≤ − .Furthermore, we determine the non-caterpillars with the smallest, the second smallest and the third smallest RCW numbers.

RCW Numbers of Non-Caterpillars
All n-vertex trees with diameter 2, 3,  .
has degree at least three.There are two cases.
has degree at least three.Let 1 2 , , , k w w w  be all the neighbors outside ( ) , where i w is a neighbor of ( ) T * be the tree formed from T by deleting edges i i w v ′ and adding edges , with equality if and only if = 2 , and ( ) . This is a contradiction.

Case 2. Any verter
, max , be the neighbors of y in T, where 1 x x = and 1 r ≥ .
Let T * * be the tree obtained from T by deleting edges i yx and adding edges , 1 , Suppose that there is a vertex , , , .

RCW T RCW T ′ >
, this is a contradiction.Thus any vertex of T outside ( ) P T has degree at most two.
Suppose that there are at least two vertices of T outside ( ) T y δ = and let x be the neighbor of y which is different from in T. Let T ′′ be the tree formed from T by deleting edge yx and adding edge This is a contradiction.Thus there is exactly one vertex outside ( ) . where is even; . where is odd Combining The result follows.Theorem 2 For 9 n ≥ , there is .

2 n − and 1 n
− are caterpillars.Let n and d be integers with of non-caterpillars with n vertices and diameter d.Let pendant vertices to one center (fixed if it is bicentral) of the path be the sum of all distances from u to the vertices in A, i.e., between vertices u and v in T.Lemma 1 Let T be a tree with minimum RCW number in

Figure 1 .
Figure 1.The tree N n,d,i .
T with degree two and all other vertices of T outside ( ) P T are pendant vertices.Then, calculation, we get( proof is finished.