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The reciprocal complementary Wiener number of a connected graph
*G* is defined as
where
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-cater- pillars with given diameter
*d*, we obtain the unique tree with minimum reciprocal complementary Wiener number, where
. We also determine the
*n*-vertex non-caterpillars with the smallest, the second smallest and the third smallest reciprocal complementary Wiener numbers.

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 [

Let G be a simple connected graph with vertex set

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 [

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

In this paper, we show that among all n-vertex non-caterpillars with given diameter d,

All n-vertex trees with diameter 2, 3,

the class of non-caterpillars obtained by attaching the stars

pendant vertices to one center (fixed if it is bicentral) of the path

Let T be a tree. For

sum of all distances from u to the vertices in A, i.e.,

Lemma 1 Let T be a tree with minimum RCW number in

Proof. Suppose that

we require that

Case 1. One of

outside

all

with equality if and only if

Then

with equality if and only if

since

Case 2. Any verter

Let

This is a contradiction.

By combining Cases 1 and 2, we find that

Lemma 2 Let

with equality if and only if

Proof. Let T be a tree with the minimum RCW number in

Suppose that there is a vertex

obtained from T by deleting edges

and then

Suppose that there are at least two vertices of T outside

with

This is a contradiction. Thus there is exactly one vertex outside

By a direct calculation, we get

Combining Lemmas 1 and 2, we get

Theorem 1 Let

with equality if and only if

Lemma 3 For

Proof. If d is even, then

If d is odd, then

The result follows.

Theorem 2 For

And

Proof. Let

and hence

Now suppose that

where equality holds if and only if

Case 1. n is odd. Let

Case 2. n is even. Let

Thus, the proof is finished.

YanliZhu,FuyiWei,FengLi, (2016) Reciprocal Complementary Wiener Numbers of Non-Caterpillars. Applied Mathematics,07,219-226. doi: 10.4236/am.2016.73020