Comparison Between Geometric-arithmetic Indices Kinkar

The concept of geometric–arithmetic indices (GA) was introduced in the chemical graph theory very recently. In this letter we compare the geometric–arithmetic indices for chemical trees, starlike trees and general trees. Moreover, we give a conjecture for general graphs. (doi: 10.5562/cca2005)


INTRODUCTION
Let G = (V, E) denote a simple graph with n vertices and m edges, V(G) = {1, 2, …, n} and m = |E(G)|. 1Also, let d i be the degree of the vertex i for i = 1, 2,…, n.The maximum vertex degree is denoted by Δ in G. Recently, a new class of topological descriptors, based on some properties of vertices of graph is presented.These indices are named as "geometric-arithmetic indices" (GA general ) and their definition is as follows: where Q i is some quantity that in a unique manner can be associated with the vertex i of the graph G.The first member of this class was considered by Vukičević and Furtula 2 by setting Q i to be the degree d i of the vertex i of the graph G: . (

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For i,jϵV(G), let d(i,j|G) be the distance between the vertices i and j in G.For ijϵE(G), The second member of this class was considered by Fath-Tabar et al. 3 by setting Q i to be the number n i of vertices of G lying closer to the vertex i than to the vertex j for the edge ij of the graph G: . ( Let x be a vertex and ij be an edge of the graph G.The distance between x and ij is defined as It should be noted that m i is not a quantity that in a unique manner can be associated with the vertex i of the graph G, but that it depends on the edge ij.Yet, this restriction is not relevant for the definition of GA 3 .Note that in all cases m i ≥ 0 and 1. Then, incorporating m i as vertex quantity into Equation (1), the third geometric-arithmetic index is defined as, 4 .

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It has been demonstrated, on the example of octane isomers, that GA index is well-correlated with a variety of physico-chemical properties. 2 Vukičević and Furtula 2 in order to study the predicive power of the GA index considered the following set of octane properties: boiling points, entropy, enthalpy of vaporization, stand-Croat.Chem.Acta 85 (2012) 353.ard enthalpy of vaporization, enthalpy of formation and acentric factor.The structure-property models based on the GA indices were comparable (and in some cases even better than) to models obtained by the connectivity index. 56][7][8][9][10][11][12][13] A survey of mathematical properties of the GA indices and their uses in QSPR and QSAR is recently given by Das, Gutman and Furtula. 14The above results indicate the potential of the GA molecular descriptors in the structure-property-activity modeling.In order to fully explore their potential, it is necessary to study the mathematical and computational properties and the range of applicability of the GA indices.The preliminary results are encouraging.We compare the first geometricarithmetic index and the atom-bond connectivity index. 15In this letter we compare the geometricarithmetic indices for chemical trees, starlike trees and general trees.Moreover, we give a conjecture for general graphs.Finally, we give conclusion.

PRELIMINARIES
A connected graph with maximum vertex degree at most 4 belongs to a family of molecular graphs depicting carbon compounds. 16Its graphical representation may resemble a structural formula of some (usually organic) molecule.That was a primary reason for employing graph theory in chemistry.Nowadays this area of mathematical chemistry is called chemical graph theory. 16A tree in which the maximum vertex degree does not exceed 4 is said to be a "chemical tree".A vertex of a graph is said to be pendent if its neighborhood contains exactly one vertex.An edge of a graph is said to be pendent if one of its vertices is a pendent vertex.Denote, as usual, by 1, 1 n K  and P n , the star and the path on n vertices, respectively.
A tree is said to be starlike if exactly one of its vertices has degree greater than two.By S(r 1 , r 1 , …, r k ), we denote the starlike tree which has a vertex 1 of degree k ≥ 3 and which has the property This tree has r 1 + r 2 + ... + r k + 1 = n vertices and assumed that r 1 ≥ r 2 ≥ ... ≥ r k ≥ 1.We say that the starlike tree S(r 1 , r 1 , ..., r k ) has k branches, the lengths of which are r 1 , r 1 , ..., r k , respectively.

COMPARISON BETWEEN GA 1 INDEX AND GA 2 INDEX
In this section we compare between GA 1 and GA 2 index for chemical trees and starlike trees.First we prove the following result: Lemma.Let T be a chemical tree.Then the number of pendent vertices in T are 2a + b + 2, where a is the number of four degree vertices and b is the number of three degree vertices in T.
Proof: If h i is the number of vertices of degree i in T, then we have From the two relations above we get where h 3 = b and h 4 = a.Hence the Lemma.

Case (a):
for each non-pendent edge ( ). ij E T  In this case there are at least three pendent edges in T. Using above results, we get GA 1 (T) > GA 2 (T) as n ≤ 9. 1), then one can see easily that GA 1 (T) > GA 2 (T).Otherwise, 7 ≤ n ≤ 9.In this case there are at least four pendent edges and at most four non-pendent edges in T as n ≤ 9. Using Table 2, we have 0.8 0.7 0.1 2 2

Case (b):
for each pendent edge ( ), ij E T  as n ≥ 7 and 0.94 1 0.0 2 2 6 for each non-pendent edge ( ). ij E T  Thus we get GA 1 (T) > GA 2 (T).This completes the proof.Now we compare between GA 1 (T) index and GA 2 (T) index for starlike trees.
Theorem 2. Let S(r 1 , r 1 , …, r k ) be a starlike tree of order n.Then with equality if and only if S is isomorphic to star K 1,n-1 .

COMPARISON BETWEEN GA 2 INDEX AND GA 3 INDEX
In this section we compare between GA 2 index and GA 3 index for any tree T. Theorem 3.For any tree T, 2 ( ) ( ).

GA T GA T 
Proof: For any tree T, for any edge ( ). ij E T  Without loss of generality, we can assume that n i ≥ n j for any non-pendent edge ( ), ij E T  that is, we have From above results we have Squaring both sides and simplifying, we get Thus we have GA 2 (T) > GA 3 (T).This completes the proof.
Corollary.Let T be a chemical tree or starlike tree of order n.Then Finally, the following conjecture holds.
Conjecture.For any connected graph G,

CONCLUSION
In this report we discuss the comparison between first and second geometric-arithmetic indices for chemical trees and starlike trees.Besides these, it has been shown that second geometric-arithmetic index is greater than to the third geometric-arithmetic index for any tree.Com-parison between these indices, in the case of general graphs, remains an open problem.
By Theorem 1, Theorem 2, and Theorem 3, we get the required result.

Table 2
 for all above degree pairs.First we assume that n ≥ 10.Let

Table 1 .
Comparison of structure-property models based on the GA indices to models obtained by the connectivity index