# New Results on the Geometric-Arithmetic Index

Authors: Akbar Jahanbani; Maryam Atapour; Zhibin Du
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetG be a graph with vertex set VG and edge set EG. Let du denote the degree of vertex u∈VG. The geometric-arithmetic index of G is defined as GAG=∑uv∈EG2dudv/du+dv. 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 several other topological indices.


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## Body

## 1. Introduction

LetG be a simple graph (i.e., graph without loops and multiple edges) with vertex set VG and edge set EG. The integers n=VG and m=EG are the order and the size of the graph G, respectively. For u∈VG, we denote by du the degree of vertex u in G. The 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 indexTopG of a graph G is a number with the property that, for every graph H isomorphic to G, TopH=TopG.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]). The 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 [4–6], 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].The geometric-arithmetic index of a graph is defined in [13] as(1)GAG=∑uv∈EG2dudvdu+dv.The geometric-arithmetic index has a number of interesting properties, e.g., see [13]. The 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 [13–16].The 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.


## 2. Preliminaries

Let us recall some remarkable lemmas which will be used in this paper.The first one is a very straightforward observation.Lemma 1 (see [17]).
Letx and y be two positive numbers. Then,(2)2xyx+y≤xy≤x+y/2+xy2≤x+y2≤x2+y22.The following is the well-known inequality of arithmetic and geometric means.Lemma 2 (inequality of arithmetic and geometric means, see [18]).
Letx1,…,xn be positive numbers. Then,(3)n1/x1+1/x2+⋯+1/xn≤∏i=1nxin≤x1+x2+⋯+xnn≤x12+x22+⋯+xn2n.Lemma 3 (see [19]).
Leta=aii=1n and bii=1n be two sequences of positive numbers. For any r≥0,(4)∑i=1nair+1br≥∑i=1nair+1∑i=1nbir.Lemma 4 (see [20]).
Letr≤ai≤R for 1≤i≤m and randR be some positive constants. Then,(5)∑i=1mai∑i=1m1ai≤m21+141−1+−1m+12m2Rr−rR2.Lemma 5 (see [21]).
Ifa1,a2,…,an and b1,b2,…,bn are positive numbers, where m1≤ai≤N1 and m2≤bi≤N2 for each 1≤i≤n, then(6)∑i=1nai2∑i=1nbi2−∑i=1naibi2≤n24N1N2+m1m2.Lemma 6 (the Pólya–Szegö inequality, see p. 62 in [22]).
Leta=aii=1n and bii=1n be two sequences of positive numbers, where 0<m1≤ai≤M1 and 0<m2≤bi≤M2, for i=1,2,…,n. Then,(7)∑i=1nai2∑i=1nbi2≤14M1M2m1m2+m1m2M1M22∑i=1naibi2.


## 3. 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 geometric-arithmetic index in terms of order, size, maximum degree, minimum degree, domination number, girth, number of cut edges, and number of pendent vertices.The first and second Zagreb indices are vertex-degree-based graph invariants defined as(8)M1G=∑uv∈EGdu+dv,M2G=∑uv∈EGdudv.The quantityM1 was first considered in 1972 [6], whereas M2 in 1975 [5]. The general Randić index is defined as follows [23]:(9)RαG=∑uv∈EGdudvα,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.
LetG be a graph. Then,(10)GAG≤M1G+2R1/2G4.Proof.
By Lemma1, we have(11)GAG=∑uv∈EG2dudvdu+dv≤∑uv∈EG2dudvdu+dv≤∑uv∈EGdu+dv/2+dudv2=∑uv∈EGdu+dv+2dudv4=M1G+2R1/2G4,as desired.
Using Lemma1 and an argument similar to the proof of Theorem 1, we can obtain the next result.Corollary 1.
LetG be a graph. Then,(12)GAG≤R1/2G.From Lemma1, we get(13)R1/2G=∑uv∈EGdudv≤∑uv∈EGdu+dv2=M1G2.Again by Lemma1, we have(14)M1G+2R1/2G4=∑uv∈EGdu+dv+2dudv4=∑uv∈EGdu+dv/2+dudv2≤∑uv∈EGdu+dv2=M1G2.Hence, we can see that the bounds in Theorem1 and Corollary 1 improve the bound:(15)GAG≤M1G2,established in [15].The proof of the following result can be found in [23].Lemma 7 (see [23]).
LetG be a graph of size m. Then,(16)RαG≤m8m+1−122α,for 0<α≤1.Using Corollary1 and Lemma 7, we can drive the next result.Corollary 2.
LetG be a graph of size m. Then,(17)GAG≤m8m+1−12.Lemma 8.
Letx and y be two positive numbers. Then,(18)2xyx+y≤1,x+yxy≥2.Now, we obtain an upper bound for the geometric-arithmetic index in terms of the first Zagreb index.Theorem 2.
LetG be a graph of order n≥2, size m, and minimum degree δ. Then,(19)GAG≤m−n+M1Gδ2.Proof.
Notice that(20)∑uv∈EGdu+dvdudv=∑uv∈EG1du+1dv=n.
By Lemma8, we have(21)GAG+n=∑uv∈EG2dudvdu+dv+du+dvdudv≤∑uv∈EG1+du+dvdudv=∑uv∈EG1+∑uv∈EGdu+dvdudv≤m+M1Gδ2,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. The domination number of a graphG is the size of a minimum dominating set.Theorem 3.
(see [24]). Let T be a tree of order n with domination number γ. Then,(22)M1T≤n−γn−γ+1+4γ−1.By Theorems2 and 3, we have the following result for trees with the given domination number.Corollary 3.
LetT be a tree of order n≥2 with domination number γ. Then,(23)GAT≤n−γn−γ+1+4γ−1−1.Since for every two real numbersx,y, and xy≤x+y2/4, we have the next observation.Lemma 9.
Letx and y be two real numbers, where x+y≠0. Then, xy/x+y2≤1/4.Next, we establish an upper bound for the geometric-arithmetic index in terms of the second Zagreb index.Theorem 4.
LetG be a graph of size m with maximum degree Δ. Then,(24)GAG≤5m4−M2G4Δ2.Proof.
By Lemmas8 and 9, we have(25)GAG+M2G4Δ2≤∑uv∈EG2dudvdu+dv+dudvdu+dv2≤∑uv∈EG2dudvdu+dv+14≤∑uv∈EG1+14=5m4,and this implies the desired bound.In [25], it is proved that, for any tree T of order n, M2T≥4n−8. Using this and Theorem 4, we obtain the next result.Corollary 4.
LetT be a tree of order n with maximum degree Δ. Then,(26)GAT≤5n−14−n−2Δ2.Here, we establish an upper bound for the geometric-arithmetic index in terms of the hyper-Zagreb index.The hyper-Zagreb index is defined as follows [9]:(27)HMG=∑uv∈EGdu+dv2.Theorem 5.
LetG be a graph of order n, size m, and minimum degree δ. Then,(28)GAG≤m−n+HMG2δ2.Proof.
By Inequality (21), we have(29)GAG+n≤∑uv∈EG1+∑uv∈EGdu+dvdudv≤∑uv∈EG1+∑uv∈EGdu+dv2dudv/du+dv=∑uv∈EG1+∑uv∈EGdu+dv22dudv≤m+HMG2δ2.
It leads to the desired bound.
The next result is proven in [26].Theorem 6 (see [26]).
LetG be a graph with n vertices and m edges. Then,(30)HMG≤m3n+1616n2n−12.
Theorems5 and 6 lead to the desired result.Corollary 5.
LetG be a graph of order n, size m, and minimum degree δ. Then,(31)GAG≤m−n+m3n+1632δ2n2n−12.The redefined third Zagreb index is defined as follows [27]:(32)ReZ3G=∑un∈EGdudvdu+dv.Now, we obtain an upper bound for the geometric-arithmetic index in terms of the second Zagreb index, the general Randić index, and the redefined third Zagreb index.Theorem 7.
LetG be a graph with maximum degree Δ and minimum degree δ. Then,(33)GAG≤M2G+R1/2Gδ−ReZ3G2Δ.Proof.
It is easy to obtain(34)M2G−GAG=∑uv∈EGdudv−2dudvdu+dv=∑uv∈EGdu+dvdudv−2dudvdu+dv=∑uv∈EGdu+dvdudvdu+dv−∑uv∈EG2dudvdu+dv≥ReZ3G2Δ−R1/2Gδ.
The desired bound follows.Theorem 8.
LetG be a graph of order n, size m, maximum degree Δ, and minimum degree δ. Then,(35)GAG≤2m2n1+141−1+−1m+12m2Δδ−δΔ2.Proof.
Now, puttingauv=2dudv/du+dv for each edge uv∈EG, R=Δ/δ, and r=δ/Δ in Lemma 4, we have(36)∑uv∈EG2dudvdu+dv∑uv∈EGdu+dv2dudv≤m21+141−1+−1m+12m2Δδ−δΔ2.
On the contrary, we have(37)n2=∑uv∈EGdu+dv2dudv≤∑uv∈EGdu+dv2dudv.
Finally, we get the bound by using Inequalities (36) and (37).
The sigma index ofG is defined in [28] as(38)σG=∑uv∈EGdu−dv2.
Here, we obtain an upper bound for the geometric-arithmetic index in terms of the first Zagreb index and the sigma index.Theorem 9.
LetG be a nontrivial graph with maximum degree Δ. Then,(39)GAG≤M1G2−σG4Δ.Proof.
For two real numbersx and y, we have that(40)xy=14x+y2−x−y2.
By (40), we obtain(41)GAG=∑uv∈EG2dudvdu+dv≤∑uv∈EG2dudvdu+dv=∑uv∈EGdu+dv2−du−dv22du+dv=12∑uv∈EGdu+dv−∑uv∈EGdu−dv22du+dv≤12∑uv∈EGdu+dv−∑uv∈EGdu−dv24Δ=M1G2−σG4Δ,and this implies the desired bound.
The general firstF-index of a graph G is defined in [29] as(42)F1aG=∑uv∈EGdu2+dv2a,where a is a real number. In particular, F11G=FG.
Since for every two real numbersx and y, x−y2≥0, and we deduce that, for any graph G,(43)FG≥2M2G,σG=FG−2M2G.
Using these and Theorem9, we obtain the next result.Corollary 6.
LetG be a nontrivial graph with maximum degree Δ. Then,(44)GAG≤M1G2−FG−2M2G4Δ.FromFG≥2M2G, we would like to indicate that the above new bound improves the known bound:(45)GAG≤M1G2,which was established in [15].Now, by using the following result, we want to obtain an upper bound for trees.Theorem 10 (see [30]).
LetT be a tree of order n with independence number α. Then,(46)M1T≤α2−3α+4n−4.Here, by Theorems9 and 10, we obtain the next result.Corollary 7.
LetT be a tree of order n with independence number α and maximum degree Δ. Then,(47)GAT≤α2−3α+4n−42−σG4Δ.


## 4. 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 geometric-arithmetic index which improve some well-known bounds.Theorem 11.
LetG be a graph of size m with minimum degree δ. Then,(48)GAG≥4δ2m2HMG.Proof.
By Lemmas1 and 2, we have(49)m2GAG=m2∑uv∈EG2dudv/du+dv≤∑uv∈EGdu+dv2dudv≤∑uv∈EGdu+dv4dudv/du+dv=∑uv∈EGdu+dv24dudv≤14δ2∑uv∈EGdu+dv2=HMG4δ2.
The result follows.Here, by Theorems11 and 6, we have the next result.Corollary 8.
LetG be a graph of order n and size m, with minimum degree δ. Then,(50)GAG≥64n2δ2n−12mn+16.Since for any real numbersx and y, it holds that x+y2/4≤x2+y2/2; hence, by this fact and Inequality (49), we can obtain the following result.Corollary 9.
LetG be a graph of size m with minimum degree δ. Then,(51)GAG≥2δ2m2FG.We start with a lower bound for the geometric-arithmetic index in terms of the generalF-index.Theorem 12.
LetG be a nontrivial graph of size m with minimum degree δ. Then,(52)GAG≥2δm2F11/2G.Proof.
Setr=1, auv=2dudv4, and buv=du2+dv2 for each uv∈EG. By Lemmas 1 and 3, we have(53)GAG=∑uv∈EG2dudvdu+dv≥∑uv∈EG2dudv2du2+dv2/2=∑uv∈EG2dudvdu2+dv2=∑uv∈EG2dudv42du2+dv2≥∑uv∈EG2dudv42∑uv∈EGdu2+dv2≥2δm2F11/2G.
The proof is completed.The harmonic index is defined as follows [11]:(54)HG=∑uv∈EG2du+dv.Theorem 13.
LetG be a nontrivial graph of order n, size m, and minimum degree δ. Then,(55)GAG≥δHG+n−2m.Proof.
Notice that(56)GAG+2m=∑uv∈EG2dudvdu+dv+∑u∈VGdu≤∑uv∈EG2dudvdu+dv+∑u∈VGδ=∑uv∈EG2dudvdu+dv+nδ≤δHG+nδ.
The result follows.
Applying (56), we obtain the next results.Corollary 10.
LetG be a nontrivial graph of order n, size m, and minimum degree δ. Then,(57)GAG≥R1/2GΔ+δn−2m.Corollary 11.
LetG be a nontrivial graph of order n, size m, and minimum degree δ. Then,(58)GAG≥δmΔ+δn−2m.Theorem 14 (see [31]).
LetG be a connected graph of order n≥3. Then,(59)HG≥2n−1n.A cut edge of a graph is an edge whose removal increases the number of connected components of the graph.Lemma 10 (see [32]).
LetG be a connected graph of order n and k′ cut edges. Then,(60)m≤n−k′n−k′−12+k′.Now, by Theorems13 and 14, and Lemma 10, we can obtain the next result.Corollary 12.
LetG be a connected graph of order n, k′ cut edges, and minimum degree δ. Then,(61)GAG≥δ2n−1n+n−2n−k′n−k′−12+k′.Here, we will use the following particular case of Jensen’s inequality.Lemma 11.
Letfx be a convex function defined in x>0. For x1,x2,…,xm>0,(62)fx1+x2+⋯+xmm≤1mfx1+fx2+⋯+fxm.The general sum-connectivity index is defined as follows [8]:(63)χαG=∑uv∈EGdu+dvα.Now, we obtain a lower bound for the geometric-arithmetic index in terms of the general sum connectivity index.Theorem 15.
LetG be a graph of size m and minimum degree δ. Then,(64)GAG≥4δ2m3χ4G.Proof.
Sincefx=1/x2 is a convex function for x>0, from Lemmas 1 and 11, we have(65)mGAG2=m∑uv∈EG2dudv/du+dv2≤1m∑uv∈EGdu+dv2dudv2≤1m∑uv∈EGdu+dv4dudv/du+dv2=1m∑uv∈EGdu+dv24dudv2≤116mδ4∑uv∈EGdu+dv4=χ4G16mδ4,as desired.Now, we obtain an upper bound for the geometric-arithmetic index in terms of the sigma index.Theorem 16.
LetG be a simple connected graph of size m with maximum degree Δ, p pendent vertices, and minimum nonpendent vertex degree δ1. Then,(66)GAG≥2pΔ1+Δ+4m−p2−m−p/δ12σG−pδ1−12Δ+δ1/2Δδ1+2Δδ1/Δ+δ1.Proof.
We partition all the edges into two parts: pendent edges and nonpendent edges, so(67)GAG=∑uv∈EGdu=12dv1+dv+∑uv∈EGdu,dv≠12dudvdu+dv.
On one hand, for the pendent edges, it is not hard to check that2dv/1+dv decreases in 2≤dv≤Δ; thus,(68)∑uv∈EGdu=12dv1+dv≥2pΔ1+Δ.
Now, we consider the nonpendent edges. It is easy to see that the functionx+1/x gets its maximum value when x attains the maximum or minimum value. From Δ/δ≥du/dv≥δ/Δ for all uandv∈VG, we have(69)dudv+dvdu≤Δδ+δΔ,which is equivalent to(70)2Δδ1Δ+δ1≤2dudvdu+dv≤1.
Setauv=1 and buv=2dudv/du+dv for each edge uv∈EG, M1=m1=M2=1, and m2=2Δδ1/Δ+δ1 in Lemma 6, and we have(71)∑uv∈EGdu,dv≠112∑uv∈EGdu,dv≠12dudvdu+dv2=m−p∑uv∈EGdu,dv≠11−du−dvdu+dv2≤1412Δδ1/Δ+δ1+2Δδ1Δ+δ12∑uv∈EGdu,dv≠12dudvdu+dv2,which implies that(72)∑uv∈EGdu,dv≠12dudvdu+dv≥4m−p∑uv∈EGdu,dv≠11−du−dv/du+dv2Δ+δ1/2Δδ1+2Δδ1/Δ+δ1≥4m−p2−m−p/δ12∑uv∈EGdu,dv≠1du−dv2Δ+δ1/2Δδ1+2Δδ1/Δ+δ1=4m−p2−m−p/δ12σG−∑uv∈EGdu=1dv−12Δ+δ1/2Δδ1+2Δδ1/Δ+δ1≥4m−p2−m−p/δ12σG−pδ1−12Δ+δ1/2Δδ1+2Δδ1/Δ+δ1.
Finally, the result follows from (67), (68), and (72).
Next, results are immediate consequences of Theorem16 with the setting p=0.Corollary 13.
For a graphG of size m with maximum degree Δ and minimum degree δ≥2,(73)GAG≥4m2−m/δ2σGΔ+δ/2Δδ+2Δδ/Δ+δ.Now, we obtain a lower bound for the geometric-arithmetic index in terms of the second Zagreb index and the general sum connectivity index.Theorem 17.
LetG be a graph of size m, maximum degree Δ, and minimum degree δ. Then,(74)GAG≥4M2Gχ−2G−m24⋅Δ2+δ2Δδ.Proof.
By Lemma5 and putting auv=2dudv, buv=1/du+dv,m1=2δ,N1=2Δ,m2=1/2Δ, and N2=1/2δ, we have(75)∑i=1n4dudv∑i=1n1du+dv2−∑i=1n2dudvdu+dv2≤m24⋅Δ2+δ2Δδ.
This implies that(76)GAG2≥4M2Gχ−2G−m24⋅Δ2+δ2Δδ.
The result follows.
Now, we obtain a lower bound for the geometric-arithmetic index in terms of the harmonic index.Theorem 18.
LetG be a graph without isolated edges. Then,(77)GAG≥2HG.Proof.
Since for eachuv∈EG, dudv≥2, we obtain(78)GAG=∑uv∈EG2dudvdu+dv≥∑uv∈EG22du+dv=2HG,as desired.
The proof of next results can be found in [33].Theorem 19 (see [33]).
LetG be a triangle-free graph of order n and the minimum degree δ≥kk≤n/2. Then,(79)HG≥2kn−kn.Theorem 20 (see [33]).
LetG be a triangle-free graph of order n and size m. Then,(80)HG≥2mn.Applying Theorems18–20, it leads to the next results.Corollary 14.
LetG be a triangle-free graph of order n without isolated edges, and the minimum degree δ≥kk≤n/2. Then,(81)GAG≥22kn−kn,(82)GAG≥22mn.We can see that Inequality (82) improves the next well-known result for triangle-free graphs [13]. Let G be a graph of order n and size m without isolated vertex. Then,(83)GAG≥2mn.The eccentricityεv of v is defined as(84)εv=maxdv,w:w∈VG,where dv,w is the length of a shortest path connecting v and w. The radius r and diameter D are defined as the minimum and maximum values among εv over all vertices v∈VG, respectively.Xu [34] showed that, for any nontrivial connected graph G of order n, size m, and radius r, HG≥m/n−r. Using this and Theorem 18, we obtain the next result.Corollary 15.
LetG be a nontrivial connected graph of order n, size m, and radius r. Then,(85)GAG≥2mn−r.Theorem 21.
LetG be a nontrivial connected graph of size m and radius r. Then,(86)GAG≥R1/2Gn−r.Proof.
Note that, for each vertexu∈VG, we have du≤n−εu. Thus, for each edge uv∈EG,(87)GAG=∑uv∈EG2dudvdu+dv≥∑uv∈EG2dudv2n−εu−εv≥∑uv∈EG2dudv2n−2r=R1/2Gn−r,as desired.Theorem 22.
LetG be a nontrivial graph of order n, size m, and p pendent edges without isolated vertex. Then,(88)GAG≥pn−1+m−pn−1−p/2.Proof.
Since0<1/duand1/dv≤1, therefore we deduce that(89)GAG=∑uv∈EG2dudvdu+dv≥∑uv∈EG1/du+1/dvdudvdu+dv=∑uv∈EG1dudv.
For each pendent edgee=uv, we clearly have 1/dudv≥1/n−1. If e=uv is a nonpendent edge, then du+dv≤2n−1−p, as any pendent vertex is adjacent to at most one of u and v. So, dudv≤du+dv/2≤n−1−p/2; hence,(90)1dudv≥1n−1−p/2.
Thus,(91)GAG≥pn−1+m−pn−1−p/2.
The desired result follows.
In [35], Kulli et al. defined the first and second generalized multiplicative Zagreb indices:(92)MZ1aG=∏uv∈EGdu+dva,MZ2aG=∏uv∈EGdudva.
Here, we obtain a lower bound in terms of the first and second generalized multiplicative Zagreb indices.Theorem 23.
LetG be a nontrivial graph of size m. Then,(93)GAG≥2mMZ21/2GMZ11Gm.Proof.
By Lemma2, we obtain(94)GAG2m=1m∑uv∈EGdudvdu+dv≥∏uv∈EGdudvdu+dvm=∏uv∈EGdudv∏uv∈EGdu+dvm=MZ21/2GMZ11Gm,as desired.Theorem 24.
LetG be a graph of size m and minimum degree δ. Then,(95)GAG≥4δ2m2HMG.Proof.
By Lemma1, we get(96)GAG2m=1m∑uv∈EGdudvdu+dv≥1m∑uv∈EG2dudv/du+dvdu+dv=1m∑uv∈EG2dudvdu+dv2≥m∑uv∈EGdu+dv2/2dudv≥m1/2δ2∑uv∈EGdu+dv2=2δ2mHMG,as desired.
In the sequel, we obtain a lower bound in terms of the first Zagreb index.Theorem 25.
LetG be a graph of size m, maximum degree Δ, and minimum degree δ. Then,(97)GAG≥δmΔ+2m−M1Gδ.Proof.
By Lemma8, we have(98)GAG+M1Gδ≥∑uv∈EG2dudvdu+dv+du+dvdudv≥∑uv∈EG2dudvdu+dv+2=∑uv∈EG2dudvdu+dv+∑uv∈EG2≥δmΔ+2m,and this implies the desired bound.Zhou [36] proved that, for any triangle-free graph of order n and size m, M1G≤mn. Together with Theorem 25, we get the next result.Corollary 16.
LetG be a triangle-free graph of order n, size m, maximum degree Δ, and minimum degree δ. Then,(99)GAG≥mδΔ+2−nδ.Inequality (98) leads to the following results.Corollary 17.
LetG be a graph of size m, maximum degree Δ, and minimum degree δ. Then,(100)GAG≥δHG+2m−M1Gδ,GAG≥R1/2GΔ+2m−M1Gδ.Note that, for every two real numbersx and y, x+y2/xy≥4. Applying this, we obtain a lower bound for the geometric-arithmetic index in terms of the hyper-Zagreb index.Theorem 26.
LetG be a graph of size m, maximum degree Δ, and minimum degree δ. Then,(101)GAG≥δmΔ+4m−HMGδ2.Proof.
From the above inequality, we have(102)GAG+HMGδ2≥∑uv∈EG2dudvdu+dv+du+dv2dudv≥∑uv∈EG2dudvdu+dv+4=∑uv∈EG2dudvdu+dv+∑uv∈EG4≥δmΔ+4m,and this implies the desired bound.
Here, we obtain a lower bound for the geometric-arithmetic index in terms of the first Zagreb index.Theorem 27.
LetG be a graph of size m and minimum degree δ. Then,(103)GAG≥2m−M1G2δ.Proof.
From the fact thatx+1/x≥2 for any x>0, we have(104)GAG+M1G2δ≥∑uv∈EG2dudvdu+dv+du+dv2dudv≥∑uv∈EG2=2m,and this implies the desired bound.Theorem 28 (see [37]).
LetG be a graph of size m and diameter D>1. Then,(105)M1G≤m2−mD−3+D−2.Now, by Theorems27 and 28, we have the following result.Corollary 18.
LetG be a graph of size m, minimum degree δ, and diameter D>1. Then,(106)GAG≥2m−m2−mD−3+D−22δ.Theorem 29 (see [38]).
LetG be a graph of size m, with t triangles and pendent vertex p. Then,(107)M1G≤mp+2+3t.Again, by Theorems27 and 29, we have the following result.Corollary 19.
LetG be a graph of size m, with t triangles, leaf number L, and minimum degree δ. Then,(108)GAG≥2m−mp+2+3t2δ.Theorem 30 (see [39]).
LetG be a triangle- and quadrangle-free graph with n>1 vertices. Then,(109)M1G≤nn−1.
Also, by Theorems27 and 30, we have the following result.Corollary 20.
LetG be a triangle- and quadrangle-free graph of order n, size m, and minimum degree δ. Then,(110)GAG≥2m−nn−12δ.



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*Source: 4901484-2021-07-31.xml*