THE TOP TEN VALUES OF HARMONIC INDEX IN CHEMICAL TREES

Let G be an n−vertex graph with degree sequence d1, d2, ..., dn. The harmonic index H(G) is defined as ) ( / G I n , where . ) / ( ∑ = n 1 i i d 1 = I(G) In this paper the top ten values of harmonic index in the set of all chem ical trees of order n are determined.


INTRODUCTION
We use West [1] for terminology and notation not defined here and consider finite simple connected graphs only.Suppose G is such a graph with V(G) = {v 1 , v 2 , …, v n }.If we sort vertices of G in such a way that deg(v 1 ) ≤ deg(v 2 ) ≤ … ≤ deg(v n ) then the sequence (d 1 , d 2 , …, d n ) is called a degree sequence for G, where A graph invariant is any function on a graph that does not depend on a labeling of its vertices.A big number of different invariants have been employed to date in chemistry for solving some chemical problems.Here we are interested to the harmonic index defined as H(B) = n/I(B), where I(B) = ( ) 1 , for a graph B. This topological index was introduced by Narumi [2].
A chemical tree is a tree in which every vertex has degree at most 4. We denote by ℊ(n), the set of all n−vertex chemical trees.It is easy to see that if A and B are two elements of ℊ(n) with the same degree sequence then H(A) = H(B).This is motivation for defining an equivalence relation ~ on ℊ(n) by A ~ B if and only if A and B have the same degree sequence.Suppose (n) denotes the set of all equivalence classes of ~ on (n) and T 1 , T 2 ∈ (n).Define T1 ≼ T2 if and only if for each element A ∈ T1 and B ∈ T2, we have H(A) ≤ H(B).
The aim of this paper is to compute the first 10 maximum value of harmonic index.We encourage the reader to consult [3−8] for basic computational techniques on the problem.

MAIN RESULTS
In this section, we are analyzing chemical trees with k th , 1 ≤ k ≤ 10, maximum values for the harmonic index.In order to formulate our results, we need introduce some graph notations used in this paper.Define: and for each i, 1≤ i ≤ r = |CT(n)|, we have: where r is the number of n−vertex chemical trees.The elements of T i,H (n) are called i th maximum class of chemical trees with respect to H index. Lemma 1.Let T n be an n−vertex chemical tree and ' n T is an n−vertex chemical tree obtained from T n by deleting a pendant vertex and appending a pendant vertex to another pendant vertex of T n .Then I( ' and for another vertex x different from u and w, ).
T T

=
On the other hand,  Therefore, and and so , 2 proving the lemma.◄ Lemma 4. Let T n be an n−vertex chemical tree containing vertices u and z of degree 3. We also assume that T 1 , T 2 and T 3 are maximal subtrees of T n with u as a pendant and z ∈ T 3 .If  In what follows ℊ(n) denotes the set of all n−vertex chemical trees.

Corollary 6.
An n−vertex chemical tree T has the maximum value of I index in ℊ(n), if T has the maximum number of pendants in ℊ(n) and non−pendant vertices satisfying r(2) + r(3) ≤ 1.
In Table 1, the number of pendants, the number of vertices of degree 2, the number of vertices of degree 3 and the number of vertices of degree 4 are computed for the maximal chemical trees with respect to I index, when the number of vertices are at most 16.In Table 2, the n−vertex chemical trees with respect to I index, 4≤ n ≤ 13, are depicted.From Table 2, it is natural to ask whether or not the maximal tree with respect to I index is unique.In the following example we respond negatively to this question.
Example 7. Consider the graphs A and B depicted in Figures 5 and 6, respectively.By simple calculations, one can see that A and B are non-isomorphic graphs with the same I index.This shows that the maximum of I index can be occurred in more than 2 chemical trees.We now compute the first ten values of harmonic index in the class of all chemical trees.At first, it is an easy fact that the path P n and the chemical tree * n P have the maximum and second maximum harmonic index, respectively., where chemical trees A δ,n are depicted in Table 3.

Corollary 2 .Figure 1 .
Figure 1.The Chemical Tree T n Containing a Subtree T 1 .

Figure 2 .
Figure 2. The Chemical Tree * n T Constructed from T 1 and T 2 .

TFigure 3 .
Figure 3.The Chemical Tree T n .Figure 4. The Chemical Tree T n ″.

Table 2 .
The Maximal Chemical Trees with respect to I Index, 4 ≤ n ≤ 13.

Figure 5 .
Figure 5.The Graph A.Figure 6.The Graph B.
harmonic index H(G) is defined as

Table 1 .
The Number of Vertices of Each Degree in n−Vertex Chemical Trees, 4 ≤ n ≤16.