# The Second Hyper-Zagreb Coindex of Chemical Graphs and Some Applications

Authors: Ahmed Ayache; Abdu Alameri; Mohammed Alsharafi; Hanan Ahmed
Journal: Journal of Chemistry
 (2021)

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## Abstract

The second hyper-Zagreb coindex is an efficient topological index that enables us to describe a molecule from its molecular graph. In this current study, we shall evaluate the second hyper-Zagreb coindex of some chemical graphs. In this study, we compute the value of the second hyper-Zagreb coindex of some chemical graph structures such as sildenafil, aspirin, and nicotine. We also present explicit formulas of the second hyper-Zagreb coindex of any graph that results from some interesting graphical operations such as tensor product, Cartesian product, composition, and strong product, and apply them on aq-multiwalled nanotorus.


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## Body

## 1. Introduction

A graph can be identified by a corresponding numerical value, a sequence of numbers, or a special polynomial or a matrix. Special attention is directed to chemical graphs which constitute a wonderful topic in graph theory because of the abundance of applications in chemistry or in medical science [1, 2]. Topological index and coindex are invariant under graph automorphism. The computation of these numerical quantities is useful and well-proven in medical information of new drugs without resorting to chemical experiments [3, 4]. All graphs in this study are finite and simple, let G be a finite simple graph on VG=n, vertices, and EG=m, edges, and the degree of a vertex v is the number of edges event to v, denoted by δGv. The complement of G, denoted by G¯, is a simple graph on the same set of vertices VG, in which two vertices u and v are adjacent by an edge uv, if and only if they are not adjacent in G. Hence, uv∈EG¯ if and only if uv∉EG. Obviously, we have EG∪​EG¯=EKn, so m¯=EG¯=n2−m, and the degree of a vertex u in G¯ is given by(1)δG¯u=n−1−δGu.Gutman and Trinajestić [5] introduced the first and second Zagreb indices as follows:(2)M1G=∑v∈VGδG2v=∑uv∈EGδGu+δGv,M2G=∑uv∈EGδGuδGv.In 2008, Došlić defined Zagreb coindices [6], which are given as follows:(3)M¯1G=∑uv∉EGδGuδGv,M¯2G=∑uv∉EGδGuδGv.Later in 2010, Ashrafi et al. have established the following nice formulas for the precise relationship between the first and second Zagreb indices and their coindices [7]:(4)M¯1G=2mn−1−M1G,M¯2G=2m2−12M1G−M2G.In 2013, Shirdel et al. [8] introduced degree-based Zagreb indices named hyper-Zagreb index which is defined as(5)HMG=∑uv∈EGδGu+δGv2.In 2013, Ranjini et al. introduced and defined the third Zagreb index of a graph as [9](6)ReZG3G=∑uv∈EGδGuδGvδGu+δGv.Furtula and Gutman in 2015 introduced the forgotten index (F-index) [10], which is defined as(7)FG=∑v∈VGδG3v=∑uv∈EGδG2u+δG2v.In 2016, De et al. introduced forgotten coindex as follows:(8)F¯G=∑uv∉EGδG2u+δG2v=n−1M1G−FG.In 2016, Veylaki et al. [11] introduced hyper-Zagreb coindex as follows:(9)HM¯G=∑uv∉EGδGu+δGv2.In 2016, Wei et al. [12] defined new version of Zagreb topological indices. It is called the hyper-Zagreb index that is defined as above. Then, the second hyper-Zagreb index of a graph G is defined as the sum of the weights δGuδGv2 and is equal to(10)HM2G=∑uv∈EGδGuδGv2.In 2020, Alameri et al. [13, 14] defined a new degree-based of Zagreb indices named Y-index and Y-coindex as(11)YG=∑uv∈EGδG3u+δG3v,Y¯G=∑uv∉EGδG3u+δG3v,where(12)Y¯G=n−1FG−YG.Here, we define a new version of Zagreb topological indices, based on the hyper-Zagreb index that is defined as above. It is called the second hyper-Zagreb index of a graphG and defined as the sum of the weights δGuδGv2, such that uv∉EG and is equal to(13)HM2¯G=∑uv∉EGδGuδGv2.Eventhough, there are several research reports contributing to the computation of topological indices of chemical graphs. However, the studies on the computation of topological coindices of octane isomers are very limited. This study focused on one of the important topological coindices named the second hyper-Zagreb coindex. Some chemical graphs were obtained by this parameter. Moreover, the second hyper-Zagreb coindex of graph operations was computed and gave some of their applications such as aq-multiwalled nanotorus.


## 2. Preliminaries

This section is devoted to some preparatory results that will play a prominent role in our study.Definition 2.1.
(see [15, 16]). Suppose that G1 and G2 are two connected graphs, then(i)
The tensor productG1⊗G2 of G1 and G2 is the graph with VG1⊗G2=VG1×VG2, and EG1⊗G2=u1,u2v1,v2|u1v1∈EG1,u2v2∈EG2.(ii)
The Cartesian productG1×G2 of G1 and G2 has the vertex set VG1×G2=VG1×VG2, and a,xb,y is an edge of G1×G2 if a = b and xy∈EG2 or ab∈EG1 and x = y.(iii)
The compositionG1G2 of G1 and G2 with disjoint vertex sets VG1 andVG2 and edge sets EG1 and EG2 is the graph with vertex set VG1×VG2 and any two verticesu=u1,v1 is adjacent with v=u2,v2 whenever (u1 is adjacent with u2) or (u1  =  u2 and v1 is adjacent with v2).(iv)
The strong productG1∗G2 of G1 and G2 is a graph with VG1∗G2=VG1×VG2, and any two vertices u1,v1 and u2,v2 are adjacent if and only if u1=u2∈VG1 andv1v2∈EG2} or v1=v2∈VG2 andu1u2∈EG1}.Lemma 2 (see [17, 18]).
LetG1 and G2 be graphs with VG1=n1, VG2=n2, EG1=m1, and EG2=m2. Then,(i)
VG1⊗G2=VG1×G2=VG1G2=VG1∗G2=n1n2(ii)
EG1⊗G2|=2m1m2(iii)
EG1×G2=m1n2+n1m2(iv)
EG1G2=m1n22+m2n1(v)
EG1∗G2=m1n2+n1m2+2m1m2(vi)
δG1⊗G2u,v=δG1uδG2v(a)
δG1×G2u,v=δG1u+δG2v(b)
δG1G2u,v=n2δG1u+δG2v(c)
δG1∗G2u,v=δG1u+δG2v+δG1uδG2vLemma 2.3.
(see [17, 18]). Let G1,G2 be two graphs with n1,n2 vertices and m1,m2 edges, respectively, then.(i)
M1G1⊗G2=M1G1M1G2(ii)
M1G1×G2=n2M1G1+n1M1G2+8m1m2(iii)
M1G1G2=n23M1G1+n1M1G2+8n2m2m1(iv)
M1G1∗G2=n2+6m2M1G1+8m2m1+6m1+n1M1G2+2M1G1M1G2Lemma 2.4.
(see [17, 18]). Let G1,G2 be two simple graphs with n1,n2 vertices and m1,m2 edges, respectively, then(i)
YG1⊗G2=YG1YG2(ii)
YG1×G2=n2YG1+n1YG2+8m1FG2+8m2FG1+6M1G1M1G2(iii)
YG1G2=n25YG1+n1YG2+8n23m2FG1+8n2m1FG2+6n22M1G1M1G2(iv)
YG1∗G2=YG14FG2+6M1G2+8m2+n2+4FG13M1G2+2m2+YG24FG1+6M1G1+8m1+n1+4FG23M1G1+2m1+YG1YG2+12FG1FG2+6M1G1M1G2Lemma 2.5.
(see [17, 18]). Let G1,G2 be two simple graphs with n1,n2 vertices and m1,m2 edges, respectively, then(i)
HM2G1⊗G2=2HM2G1HM2G2(ii)
HM2G1×G2=n2HM2G1+n1HM2G2+3FG1M1G2+3FG2M1G1+m1YG2+4ReZG3G2+m2YG1+4ReZG3G1+4M1G1M2G2+4M1G2M2G1(iii)
HM2G1G2=n26HM2G1+n1HM2G2+n24m2YG1+4ReZG3G1+4n2m1ReZG3G2+3n22FG1M1G2+n22M1G1FG2+4M2G2+m1M12G2+4n2m24n2m2M2G1+M1G1M1G2(iv)
HM2G1∗G2=HM2G2n1+10m1+10m1G1+8m2G1+6FG1+4ReZG3G1+YG1+HM2G2n2+10m2+10m1G2+8m2G2+6FG2+4ReZG3G2+YG2+YG2m1+2M1G1+4M2G1+FG1+2ReZG3G1YG1m2+2M1G2+4M2G2+FG2+2ReZG3G2+4ReZG3G2m1+2M1G1+2M2G1+2FG1+4ReZG3G1m2+2M1G2+2M2G2+2FG2+FG23M1G1+8M2G1+4M1G1M2G2+4M1G2M2G1+2HM2G1HM2G2+5FG1FG2+6ReZG3G1ReZG3G2


## 3. Main Results

In the following section, we study the second hyper-Zagreb coindex of some chemical graph structures, exactly sildenafil, aspirin, and nicotine.Proposition 3.1.
LetG be a graph with n vertices and m edges. Then,(14)HM¯G=n−2M1G+4m2−HMG.Proof.
For the proof (Theorem 3.2), we refer to [10].Proposition 3.2.
LetG be a graph with n vertices and m edges. Then,(15)HM2¯G=12M12G−12YG−HM2G.Proof.
By definition of the second hyper-Zagreb coindex and using a similar method, as above in Proposition3.1, then(16)HM2G+HM2¯G=∑uv∈EG+∑uv∉EGδGuδGv2=12∑u∈VG∑v∈VGδGuδGv2−∑v∈VGδGvδGv2=12M12G−YG.
Sildenafil (C22H30N6O4S) is a drug used for pulmonary arterial hypertension. It is taken by mouth or injection into a vein (Figure 1) [19].Figure 1
Graph structure of sildenafil.Proposition 3.3.
The second hyper-Zagreb coindex of sildenafil.From the graph structure of sildenafil (Figure1), it is easy to obtain the dataset in Tables 1 and 2.By Table1 and definitions of the first Zagreb index and the Y-index, we have(17)M1sildenafil=71+144+119+116=178,Ysildenafil=71+1416+1181+1256=1378.Also, by Table2 and definition of the second hyper-Zagreb index, we have(18)HM2sildenafil=24+39+716+1636+681+2144=1497.Using Proposition3.2, we have(19)HM2¯sildenafil=13656.Aspirin (C9H8O4) is known as acetylsalicylic acid (ASA). Aspirin has many medicinal uses as it is a drug that is used to reduce fever or inflammation, also given after a heart attack to reduce the risk of death. Aspirin is also used as a nonsteroidal anti-inflammatory drug because it has an antiplatelet effect by inhibiting its normal functioning. Also, a lot of evidence indicates that aspirin is considered a chemical agent that may limit and reduce the incidence of general cancers (Figure 2) [20, 21].Table 1
Atoms dataset of the graph structure of sildenafil.
No. of atoms714111δv1234δ2v14916δ4v11681256Table 2
Links dataset of the graph structure of sildenafil.
No. of links2371662δ2uδ2v49163681144Figure 2
Graph structure of aspirin.Proposition 3.4.
The second hyper-Zagreb coindex of aspirin.From the graph structure of aspirin (Figure2), it is easy to obtain the dataset in Tables 2 and 3.Also, by Table4 and definition of the second hyper-Zagreb index, we have(20)HM2aspirin=49+316+436+281=390.Using Proposition3.2, we have(21)HM2¯aspirin=1206.Nicotine (C10H14N2) is an alkaloid that is widely used as an anxiolytic. Nicotine is used as a drug to quit smoking, and if it is not used well, it can lead to addiction. Many types of research conducted on animals indicate that some inhibitors found in tobacco smoke, such as monoamine oxidase, may enhance some of the addictive properties of nicotine (Figure 3) [21, 22]. Any unexplained terminology is standard, typically as in [22–24].Table 3
Atoms dataset of the graph structure of aspirin.
No. of atoms454δv123δ2v149δ4v11681Table 4
Links dataset of the graph structure of aspirin.
No. of links4342δ2uδ2v9163681By Table3 and definitions of the first Zagreb index and the Y-index, we have(22)M1aspirin=41+54+49=60,Yaspirin=41+516+481=408.Figure 3
Graph structure of nicotine.Proposition 3.5.
The second hyper-Zagreb coindex of nicotine.From the graph structure of nicotine (Figure3), it is easy to obtain the dataset in Tables 5 and 6.By Table5 and definitions of the first Zagreb index and the Y-index, we have(23)M1nicotine=11+84+39=60,Ynicotine=11+816+381=372.Also, by Table6 and definition of the second hyper-Zagreb index, we have(24)HM2nicotine=19+616+436+281=483.Using Proposition3.2, we have(25)HM2¯nicotine=1131.Table 5
Atoms dataset of the graph structure of nicotine.
No. of atoms183δv123δ2v149δ4v11681Table 6
Links dataset of the graph structure of nicotine.
No. of links1642δ2uδ2v9163681


## 4. Applications

In the following section, we provide the exact value of the second hyper-Zagreb coindex of graphs that are arisen from mathematical operations such as the tensor productG1⊗G2, the Cartesian product G1×G2, the composition G1G2, and the strong product G1∗G2. Also, we apply this coindex on a q-multiwalled nanotorus.Theorem 4.1.
The second hyper-Zagreb coindex ofG1×G2 is given by(26)HM2¯G1×G2=12n2M1G1+n1M1G2+8m1m22−12n2YG1+n1YG2+8m1FG2+8m2FG1+6M1G1M1G2−n2HM2G1+n1HM2G2+3FG1M1G2+3FG2M1G1+m1YG2+4ReZG3G2+m2YG1+4ReZG3G1+4M1G1M2G2+4M1G2M2G1.Proof.
We haveHM2¯G=1/2M12G−1/2YG−HM2G, given in Proposition 3.2, and by replacing each G by G1×G2, which yields HM2¯G1×G2=1/2M12G1×G2−1/2YG1×G2−HM2G1×G2, and by using (Lemma 2.2–Lemma 2.4), we obtain the required.
All proofs in Theorems4.2–4.4 are given as Theorem 4.1.Theorem 4.2.
The second hyper-Zagreb coindex ofG1∗G2 is given by(27)HM¯2G1∗G2=12n2+6m2M1G1+8m2m16m1+n1M1G2+2M1G1M1G22−12YG14FG2+6M1G2+8m2+n2+4FG13M1G2+2m2+YG24FG1+6M1G1+8m1+n1+4FG23M1G1+2m1+YG1YG2+12FG1FG2+6M1G1M1G2−HM2G2n1+10m1+10M1G1+8M2G1+6FG1+4ReZG3G1+YG1+HM2G1n2+10m2+10M1G2+8M2G2+6FG2+4ReZG3G2+YG2+YG2m1+2M1G1+4M2G1+FG1+2ReZG3G1+YG1m2+2M1G2+4M2G2+FG2+2ReZG3G2+4ReZG3G2m1+2M1G1+2M2G1+2FG1.+4ReZG3G1m2+2M1G2+2M2G2+2FG2+FG23M1G1+8M2G1+FG13M1G2+8M2G2.Theorem 4.3.
The second hyper-Zagreb coindex ofG1⊗G2 is given by(28)HM¯2G1⊗G2=12M1G1M1G22−12YG1YG2−2HM2G1HM2G2.Theorem 4.4.
The second hyper-Zagreb coindex ofG1G2 is given byProof.
HM2¯G1G2=1/2n23M1G1+n1M1G2+8n2m2m1YG1G2=n25YG1+n1YG2+8n23m2FG1+8n2m1FG2+6n22M1G1M1G22+−1/2n25YG1+n1YG2+8n23m2FG1+8n2m1FG2+6n22M1G1M1G2−n26HM2G1+n1HM2G2+n24m2YG1+4ReZG3G1+4n2m1ReZG3G2+3n22FG1M1G2+n22M1G1FG2+4M2G2+m1M12G2+4n2m24n2m2M2G1+M1G1M1G2.
In [19, 25–27], authors computed some topological indices of molecular graph of a nanotorus (Figure 4). In this section, we compute the second hyper-Zagreb coindex of a molecular graph of a nanotorus.Figure 4
The molecular graph of a nanotorus.Corollary 4.5.
LetT = T[p; q] be the molecular graph of a nanotorus. Then, the first Zagreb index of a q-multiwalled nanotorus is M1Pn×T=pq25n−18.Proof.
The proof of the above corollary is given by Gao et al. in [3]. Obviously,(29)VG1=VPn=n,EG1=EPn=n−1,VG2=VT=pq,EG2=ET=32pq,M1G1=M1Pn=4n−6,M1G2=M1T=9pq.Corollary 4.6.
LetT = T[p; q] be the molecular graph of a nanotorus. Then, the Y-index of a q-multiwalled nanotorus is(30)YPn×T=pq625n−738.Proof.
We have by Lemma 4.2,(31)YG1×G2=n2YG1+n1YG2+8m1FG2+8m2FG1+6M1G1M1G2.
Then,(32)YPn×T=pqYPn+nYT+8n−1FT+832pqFPn+6M1PnM1T.
Therefore,(33)YPn×T=pq16n−30+n81pq+8n−127pq+12pq8n−14+64n−69pq=pq625n−738.Corollary 4.7.
LetT = T[p; q] be the molecular graph of a nanotorus. Then, the second hyper-Zagreb index of a q-multiwalled nanotorus is(34)HM2Pn×T=12pq2561n−3632.Proof.
We have by Lemma2.5,(35)HM2G1×G2=n2HM2G1+n1HM2G2+3FG1M1G2+3FG2M1G1+m1YG2+4ReZG3G2+m2YG1+4ReZG3G1+4M1G1M2G2+4M1G2M2G1.
As proof in Corollary4.6, we have(36)HM2Pn×T=12pq2561n−3632,where(37)HM2Pn=16n−40,HM2T=2432pq.
Now, we apply the second hyper-Zagreb coindex on aq-multiwalled nanotorus using Cartesian product operation.Corollary 4.8.
LetT = T[p; q] be the molecular graph of a nanotorus. Then, the second hyper-Zagreb coindex of a q-multiwalled nanotorus is(38)HM2¯Pn×T=12pqpq25n−182−3186n+4370.Proof.
We have by Proposition3.2,(39)HM2¯G=12M12G−12YG−HM2G.
Then,(40)HM2¯Pn×T=12M12Pn×T−12YG−HM2Pn×T.
By usingCorollaries 4.5–4.7, we obtain(41)HM2¯Pn×T=12pqpq25n−182−3186n+4370.


## 5. Conclusion

In this study, we obtained the value of the second hyper-Zagreb coindex of some chemical graphs, and we computed some explicit formulas for their numbers under several graph operations. Also, we applied the second hyper-Zagreb coindex on aq-multiwalled nanotorus. The results of this work may be used as a predictor, especially in the chemical graph theory. For example, in quantitative structure-activity relationships (QSAR) modelling, the predictors consist of theoretical molecular descriptors of chemicals.



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