Comparison Between Zagreb Eccentricity Indices and the Eccentric Connectivity Index , the Second Geometric-arithmetic Index and the Graovac-Ghorbani Index

The concept of Zagreb eccentricity indices ( 1 E and 2 E ) was introduced in the chemical graph theory very recently. The eccentric connectivity index ( ) c ξ is a distance-based molecular structure descriptor that was used for mathematical modeling of biological activities of diverse nature. The second geometric-arithmetic index 2 ( ) GA was introduced in 2010, is found to be useful tool in QSPR and QSAR studies. In 2010 Graovac and Ghorbani introduced a distance-based analog of the atom-bond connectivity index, the Graovac-Ghorbani index ( ) GG ABC , which yielded promising results when compared to analogous descriptors. In this note we prove that  1( ) ( ) c E T ξ T for chemical trees T. For connected graph G of order n with maximum degree Δ , it is proved that  2 ( ) ( ) c ξ G E G if   Δ 1 n and  2 ( ) ( ) c ξ G E G , otherwise. Moreover, we show that  2 GG GA ABC for paths and some class of bipartite graphs.


INTRODUCTION
topological index is a numerical descriptor of the molecular structure derived from the corresponding molecular graph.There are numerous topological descriptors that have found some applications in theoretical chemistry, especially in QSPR/QSAR research. [1,2]They can be classified based on the structural properties of graphs used for their calculation.The following topological indices are well-studied by the researchers: Wiener index, [3] Hosoya index, [4] the energy [5] and the Randić connectivity index. [6]et ( , ) G V E  denote a simple graph with n vertices and m edges, where S , is the number of elements in S).The degree of a vertex ( ) d v is the number of edges incident to i v .The maximum degree of a graph G is denoted by Δ , that is, Δ max{ ( ): ( )} . The distance between i v and j v in ( ) V G , ( , ) , is the length of a shortest i v to j v path in G.The eccentricity, ( ) v and any other vertex in G, that is, ( ) max{ ( , ) : . The diameter of G, d, is defined as the maximum value of the eccentricities of the vertices of G, that is, max{ ( , ) : , ( )} Gutman and Trinajstić [7] derived a formula for estimating total π -electron energy of conjugated systems.Their formula contained two terms that later became known as the Zagreb indices 1  M and 2 M .The first Zagreb index 1 ( ) M G and the second Zagreb index 2 ( ) M G of graph G (see Refs. [2], [7-11] and the references therein) are among the oldest and most studied topological indices.They are defined as: where ( )

A
The invariants based on vertex eccentricities attracted some attention in chemistry.In an analogy with the first and the second Zagreb indices, Ghorbani et al. [12] and Vukičević et al. [13] defined the first 1 E , and the second 2 E , Zagreb eccentricity indices by where ( ) Upper and lower bounds for the Zagreb eccentricity indices of graphs have been reported in Refs.[12-15].
The eccentric connectivity index of a graph G, denoted by ( ) C ξ G , is defined as [16] ( ) ( ) ε v are the degree and the eccentricity of the vertex i v in G, respectively.The eccentric connectivity index provides good correlations with regard to both physical and biological properties. [17]The simplicity amalgamated with high correlating ability of this index can be easily exploited in QSPR/QSAR studies.Such studies can easily provide valuable leads for the development of potential therapeutic agents.We encourage the reader to consult papers [18,19] for the mathematical properties of the eccentric connectivity index.
Let e be an edge of the graph G (which may contain cycles or be acyclic), connecting the vertices i v and j v .

Here we define two sets ( | )
i N e G and ( | ) j N e G as follows: The number of elements of ( | ) Recently, Fath-Tabar, Furtula and Gutman [20] defined second geometric-arithmetic index by For the mathematical properties of 2 GA index, the reader is referred to Refs.[20-22]  In Ref. [23], Graovac and Ghorbani proposed the following distance-based analog of the ABC index: ( )
Some initial studies indicate that the Graovac-Ghorbani index could be an effective predictive tool in chemistry.For instance, it can be used to model both the boiling and the melting points of molecules. [24]Upper and lower bounds for the GG ABC index of graphs have been given in Refs.[23], [25-28].
Let  be the class of finite graphs.A topological index is a function  M for chemical trees, molecular graphs and some graph families.Several relations between the two ABC-indices are established.Geometric-arithmetic indices are compared for chemical trees, starlike trees and general trees in Ref. [30], and the Wiener index and the Zagreb indices and the eccentric connectivity index for trees in Ref. [31].In this note we prove that 1 ( ) ( ) for chemical trees T, and for connected graph G, for paths and some class of bipartite graphs.

PRELIMINARIES
A connected graph with maximum vertex degree at most 4 is said to be a "molecular graph". [1]A tree in which the maximum vertex degree does not exceed 4 is said to be a "chemical tree".Denote, as usual, by 1, 1 n K  , n P , n C and n K , the star, the path, the cycle and the complete graph on n vertices, respectively.A double star of order n, denoted by ( , ) DS p q ( , 2) p q n p q     , is a tree, which is constructed by joining the central vertices of two stars 1, p K and 1,q K .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.

COMPARISON BETWEEN E1 AND ξ C OF GRAPHS
In this section we compare the first Zagreb eccentricity index 1 ( ) E and the eccentric connectivity index ( ) C ξ for graphs.For . Therefore the first Zagreb eccentricity index and the eccentric connectivity index are incomparable on the class of general graphs.But we have the following theorem.
Theorem 3.1.Let T be a chemical tree of order . For 4 d  , the number of non-pendent vertices in T is at most five and the number of pendent vertices in T is at least two (since T is a chemical tree).Exactly one non-pendent vertex, say i v , has eccentricity 2 and all the other non-pendent vertices have eccentricity exactly 3.For each non-pendent vertex ( ) .
All the other pendent vertices ( ) Since the number of non-pendent vertices is at most five and the number of pendent vertices is at least two in T, we have For d = 5 or 6, there are at most two vertices of eccentricity 3 with 2 ( ) ( ) ( ) 3 and there are at least two pendent vertices of eccentricity 5 with 2 ( ) ( ) ( ) 20 . For all other vertices ( ) This completes the proof of the theorem.

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In Ref. [29], we compared 1 M and c ξ for chemical tree and molecular graph.Moreover, we compare 1 E and c ξ for chemical tree in Theorem 3.1.We now obtain the following result for any connected graph.
Theorem 3.2.Let G be a connected graph.Then with equality holding if and only if ( ) ( ) . This completes the proof.
This completes the proof of the theorem.

COMPARISON BETWEEN ABCGG AND GA2 OF GRAPHS
Since each term of .
From these examples, we can conclude that 2 GA index and GG ABC index are incomparable on the class of general graphs on n vertices.So now we compare these two indices for special class of graphs.For path 1 2 1 : , and . 2 Therefore there is a term in

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We are now ready to give the proof of Let us consider a function Then by Lemma 5.1, we have ( ) (1) ( 1) 1 ( 2). 4  4 From the above, one can easily see that  , from Eq. ( 5),Eq (6) and the above result, we have The expression ( 7) is certainly non-negative if for every ( ) This completes the proof of the theorem.

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Remark 5.4.We can construct several graphs such that the condition in Theorem 5.3 is satisfied.For 100 n  , we have

CONCLUSION
Topological indices are graph invariants and are used for quantitative structure -activity relationship (QSAR) and quantitative structure -property relationship (QSPR) studies.Many topological indices have been defined in the literature and several of them have found applications as means to model physical, chemical, pharmaceutical and other properties of molecules.The eccentric connectivity index provides good correlations with regard to both physical and biological properties.In this note we presented that the eccentric connectivity index ( C ξ ) is less than the first Zagreb eccentricity index 1 ( ) E for chemical trees.For connected graph G, we prove that of a set S, denoted | | counts the number of vertices of G lying closer to the vertex i v than to vertex j v .The meaning of ( | ) j n e G is analogous.Vertices equidistant from both ends of the edge i j v v belong neither to ( | ) .Note that for any edge e of G, sake of brevity, if there is no risk of confusion, we always simplify ( | )

2
Das and Trinajstić compared the first geometricarithmetic index and the atom-bond connectivity index for trees and graphs.Moreover, they compared c ξ with 1 M and 2

Theorem 4 . 1 .
Let G be a connected graph of order 1 n  with maximum degree Δ is interesting to compare of these two indices.We start with some examples: Example 1.

2
The Graovac-Ghorbani index is a distance-based analog of the atombond connectivity index, one of the most meaningful degree-based molecular structure descriptors.In this work, we show that the second geometric-arithmetic index 2 ( ) GA is greater than the Graovac-Ghorbani index ( ) GG ABC for paths and some class of bipartite graphs.There are many unsolved problems regarding the comparison between topological indices of graphs.The comparison between the first Zagreb eccentricity index 1 ( ) E and the eccentric connectivity index ( ) C ξ , in the case of trees and general graphs is left as an open problem.The comparison between the second geometric-arithmetic index 2 ( ) GA and the Graovac-Ghorbani index ( ) GG ABC for general graphs remains a task for future.
Top from  into real numbers, where 1 Top and 2 Top .Since 1 Top and 2 Top are real numbers for any graph G, then it is interesting to compare these two topological indices 1 Top and 2