Research Article New Results on the Geometric-Arithmetic Index

Let G be a graph with vertex set V ( G ) and edge set E ( G ) . Let d u denote the degree of vertex u ∈ V ( G ) . The geometric-arithmetic index of G is deﬁned as GA ( G ) � 􏽐 uv ∈ E ( G ) ( 2 ���� d u d v 􏽰 / ( d u + d v )) . In this paper, we obtain some new lower and upper bounds for the geometric-arithmetic index and improve some known bounds. Moreover, we investigate the relationships between geometric-arithmetic index and


Introduction
Let G be a simple graph (i.e., graph without loops and multiple edges) with vertex set V(G) and edge set E(G). e integers n � |V(G)| and m � |E(G)| are the order and the size of the graph G, respectively. For u ∈ V(G), we denote by d u the degree of vertex u in G. e minimum and maximum degrees of a graph are denoted by δ and Δ, respectively.
Graph theory has provided chemists with a variety of useful tools, such as topological indices. A topological index Top(G) of a graph G is a number with the property that, for every graph H isomorphic to G, Top(H) � Top(G).
Molecular descriptors play a significant role in mathematical chemistry, especially in QSPR/QSAR investigations. Among them, special place is reserved for so-called topological descriptors. A topological index is a numeric quantity from the structural graph of a molecule.
Usage of topological indices in chemistry began in 1947 when Wiener [1] developed the most widely known topological descriptor, namely, the Wiener index, and used it to determine physical properties of types of alkanes known as paraffin (see, for instance, [2,3]). e interest of topological indices lies in the fact that they synthesize some of the properties of a molecule into a single number. With this in mind, hundreds of topological indices have been introduced and studied. Topological indices based on the vertex degree play a vital role in mathematical chemistry, and some of them are recognized as tools in chemical research.
Authors are studying various topological descriptors, such as Zagreb indices [4][5][6], general sum-connectivity index [7,8], hyper-Zagreb index [9], and harmonic index [10,11]. Besides the abovementioned ones, there are other topological descriptors based on end vertex degrees of edges of graphs that have found some applications in QSPR/QSAR research [2,12,13]. e geometric-arithmetic index of a graph is defined in [13] as e geometric-arithmetic index has a number of interesting properties, e.g., see [13]. e lower and upper bounds of the geometric-arithmetic index of connected graphs and the characterizations of graphs for which these bounds are best possible can be found in [13][14][15][16]. e aim of this paper is to investigate new relationships between the geometric-arithmetic index and other topological indices. In particular, we obtain some lower and upper bounds for the geometric-arithmetic index. Moreover, we improve some known bounds.

Preliminaries
Let us recall some remarkable lemmas which will be used in this paper. e first one is a very straightforward observation.
Lemma 1 (see [17]). Let x and y be two positive numbers. en, (2) e following is the well-known inequality of arithmetic and geometric means.
Lemma 2 (inequality of arithmetic and geometric means, see [18]). Let x 1 , . . . , x n be positive numbers. en, Lemma 3 (see [19]). Let a � (a i ) n i�1 and (b i ) n i�1 be two sequences of positive numbers. For any r ≥ 0, Lemma 4 (see [20]). Let r ≤ a i ≤ R for 1 ≤ i ≤ m and r and R be some positive constants. en, Lemma 5 (see [21]). If a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . , b n are positive numbers, where m 1 ≤ a i ≤ N 1 and Lemma 6 (the Pólya-Szegö inequality, see p. 62 in [22]). Let a � (a i ) n i�1 and (b i ) n i�1 be two sequences of positive numbers,

Upper Bounds for the Geometric-Arithmetic Index
In this section, we investigate the relationships between geometric-arithmetic index and some topological indices. Moreover, we obtain some upper bounds for the geometricarithmetic index in terms of order, size, maximum degree, minimum degree, domination number, girth, number of cut edges, and number of pendent vertices. e first and second Zagreb indices are vertex-degreebased graph invariants defined as e quantity M 1 was first considered in 1972 [6], whereas M 2 in 1975 [5]. e general Randić index is defined as follows [23]: where α is a real number. We begin with the establishment of an upper bound for the geometric-arithmetic index in terms of the first Zagreb index and the general Randić index. Theorem 1. Let G be a graph. en, Proof. By Lemma 1, we have as desired.
From Lemma 1, we get Again by Lemma 1, we have Hence, we can see that the bounds in eorem 1 and Corollary 1 improve the bound: established in [15]. e proof of the following result can be found in [23].
Using Corollary 1 and Lemma 7, we can drive the next result.

Corollary 2. Let G be a graph of size m. en,
Lemma 8. Let x and y be two positive numbers. en, Now, we obtain an upper bound for the geometricarithmetic index in terms of the first Zagreb index.

Theorem 2.
Let G be a graph of order n ≥ 2, size m, and minimum degree δ. en, Proof. Notice that By Lemma 8, we have and this implies the desired bound.
A dominating set of a graph is a vertex subset whose closed neighborhood includes all vertices of the graph. e domination number of a graph G is the size of a minimum dominating set. □ Theorem 3 (see [24]). Let T be a tree of order n with domination number c. en, By eorems 2 and 3, we have the following result for trees with the given domination number.

Corollary 3.
Let T be a tree of order n ≥ 2 with domination number c. en, Since for every two real numbers x, y, and xy ≤ ((x + y) 2 /4), we have the next observation.

Lemma 9. Let x and y be two real numbers
Next, we establish an upper bound for the geometricarithmetic index in terms of the second Zagreb index.
Journal of Mathematics Proof. By Lemmas 8 and 9, we have and this implies the desired bound.

□
In [25], it is proved that, for any tree T of order n, M 2 (T) ≥ 4n − 8. Using this and eorem 4, we obtain the next result.

Corollary 4.
Let T be a tree of order n with maximum degree Δ. en, Here, we establish an upper bound for the geometricarithmetic index in terms of the hyper-Zagreb index. e hyper-Zagreb index is defined as follows [9]: Theorem 5. Let G be a graph of order n, size m, and minimum degree δ. en, Proof. By Inequality (21), we have It leads to the desired bound. e next result is proven in [26].
□ Theorem 6 (see [26]). Let G be a graph with n vertices and m edges. en, eorems 5 and 6 lead to the desired result.
Corollary 5. Let G be a graph of order n, size m, and minimum degree δ. en, e redefined third Zagreb index is defined as follows [27]: Now, we obtain an upper bound for the geometricarithmetic index in terms of the second Zagreb index, the general Randić index, and the redefined third Zagreb index.
Theorem 7. Let G be a graph with maximum degree Δ and minimum degree δ. en, Proof. It is easy to obtain e desired bound follows.
□ Theorem 8. Let G be a graph of order n, size m, maximum degree Δ, and minimum degree δ. en, Proof. Now, putting a uv � (2 Journal of Mathematics On the contrary, we have Finally, we get the bound by using Inequalities (36) and (37). e sigma index of G is defined in [28] as Here, we obtain an upper bound for the geometricarithmetic index in terms of the first Zagreb index and the sigma index. □ Theorem 9. Let G be a nontrivial graph with maximum degree Δ. en, Proof. For two real numbers x and y, we have that By (40), we obtain and this implies the desired bound. e general first F-index of a graph G is defined in [29] as where a is a real number. In particular, F 1 1 (G) � F(G). Since for every two real numbers x and y, (x − y) 2 ≥ 0, and we deduce that, for any graph G, (43) Using these and eorem 9, we obtain the next result. □ Corollary 6. Let G be a nontrivial graph with maximum degree Δ. en, From F(G) ≥ 2M 2 (G), we would like to indicate that the above new bound improves the known bound: which was established in [15]. Now, by using the following result, we want to obtain an upper bound for trees.
Theorem 10 (see [30]). Let T be a tree of order n with independence number α. en, Here, by eorems 9 and 10, we obtain the next result.

Corollary 7.
Let T be a tree of order n with independence number α and maximum degree Δ. en,

Lower Bounds for the Geometric-Arithmetic Index
In this section, we first investigate the relationships between the geometric-arithmetic index and some other topological indices, and then, we obtain some lower bounds for the geometricarithmetic index which improve some well-known bounds.
Theorem 11. Let G be a graph of size m with minimum degree δ. en, Proof. By Lemmas 1 and 2, we have Journal of Mathematics e result follows.

Corollary 9.
Let G be a graph of size m with minimum degree δ. en, We start with a lower bound for the geometric-arithmetic index in terms of the general F-index. Theorem 12. Let G be a nontrivial graph of size m with minimum degree δ. en, Proof. Set r � 1, a uv � ����� 2d u d v 4 , and b uv � ������ d 2 u + d 2 v for each uv ∈ E(G). By Lemmas 1 and 3, we have .
□ e harmonic index is defined as follows [11]: Theorem 13. Let G be a nontrivial graph of order n, size m, and minimum degree δ. en, Proof. Notice that e result follows. Applying (56), we obtain the next results.
Corollary 11. Let G be a nontrivial graph of order n, size m, and minimum degree δ. en, Theorem 14 (see [31]). Let G be a connected graph of order n ≥ 3. en, A cut edge of a graph is an edge whose removal increases the number of connected components of the graph.
Lemma 10 (see [32]). Let G be a connected graph of order n and k ′ cut edges. en, Now, by eorems 13 and 14, and Lemma 10, we can obtain the next result.

Corollary 12.
Let G be a connected graph of order n, k ′ cut edges, and minimum degree δ. en, Here, we will use the following particular case of Jensen's inequality.

Lemma 11. Let f(x) be a convex function defined in
(62) e general sum-connectivity index is defined as follows [8]: Now, we obtain a lower bound for the geometric-arithmetic index in terms of the general sum connectivity index.
Theorem 15. Let G be a graph of size m and minimum degree δ. en, Proof. Since f(x) � (1/x 2 ) is a convex function for x > 0, from Lemmas 1 and 11, we have as desired.
□ Now, we obtain an upper bound for the geometricarithmetic index in terms of the sigma index.
Theorem 16. Let G be a simple connected graph of size m with maximum degree Δ, p pendent vertices, and minimum nonpendent vertex degree δ 1 . en, Proof. We partition all the edges into two parts: pendent edges and nonpendent edges, so On one hand, for the pendent edges, it is not hard to check that Now, we consider the nonpendent edges. It is easy to see that the function x + (1/x) gets its maximum value when x attains the maximum or minimum value. From Journal of Mathematics 7 which is equivalent to Set a uv � 1 and b uv � (2 for each edge uv ∈ E(G), M 1 � m 1 � M 2 � 1, and m 2 � (2 ��� � Δδ 1 /Δ + δ 1 ) in Lemma 6, and we have which implies that Now, we obtain a lower bound for the geometricarithmetic index in terms of the second Zagreb index and the general sum connectivity index.
Theorem 17. Let G be a graph of size m, maximum degree Δ, and minimum degree δ. en, and is implies that e result follows. Now, we obtain a lower bound for the geometricarithmetic index in terms of the harmonic index. □ Theorem 18. Let G be a graph without isolated edges. en, Proof. Since for each uv ∈ E(G), d u d v ≥ 2, we obtain as desired. e proof of next results can be found in [33].
□ Theorem 19 (see [33]). Let G be a triangle-free graph of order n and the minimum degree δ ≥ k(k ≤ (n/2)). en, Theorem 20 (see [33]). Let G be a triangle-free graph of order n and size m. en, Applying eorems 18-20, it leads to the next results.

Corollary 14.
Let G be a triangle-free graph of order n without isolated edges, and the minimum degree δ ≥ k(k ≤ (n/2)). en, We can see that Inequality (82) improves the next wellknown result for triangle-free graphs [13]. Let G be a graph of order n and size m without isolated vertex. en, GA(G) ≥ 2m n .
(83) e eccentricity ε(v) of v is defined as where d(v, w) is the length of a shortest path connecting v and w. e radius r and diameter D are defined as the minimum and maximum values among ε(v) over all vertices v ∈ V(G), respectively.
Xu [34] showed that, for any nontrivial connected graph G of order n, size m, and radius r, H(G) ≥ (m/n − r). Using this and eorem 18, we obtain the next result.
Corollary 15. Let G be a nontrivial connected graph of order n, size m, and radius r. en, Theorem 21. Let G be a nontrivial connected graph of size m and radius r. en, Proof. Note that, for each vertex u ∈ V(G), we have d u ≤ n − ε(u). us, for each edge uv ∈ E(G), as desired.