# Computation of Vertex Degree-Based Molecular Descriptors of Hydrocarbon Structure

Authors: Zeeshan Saleem Mufti; Rukhshanda Anjum; Ayesha Abbas; Shahbaz Ali; Muhammad Afzal; Ashraful Alam
Journal: Journal of Chemistry
 (2022)

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## Abstract

Topological indices are such numbers or set of numbers that describe topology of structures. Nearly 400 topological indices are calculated so far. The prognostication of physical, chemical, and biological attributes of organic compounds is an important and still unsolved problem of computational chemistry. Topological index is the tool to predict the physicochemical properties such as boiling point, melting point, density, viscosity, and polarity of organic compounds. In this study, some degree-based molecular descriptors of hydrocarbon structure are calculated.


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## Body

## 1. Introduction

The invention of graph theory in 18th century was a biggest game changer in the field of mathematics by a Swiss Mathematician Leonard Euler (1702–1782). He used graphs to tackle the famous problem of Konigsberg bridge [1, 2]. In this study, G=V,E is a simple undirected graph containing a set of vertices V and an edges set E [3]. The number of lines connected to a vertex is called a degree of a vertex and is denoted by Degu.Topological indices investigate the features of graphs that persist constant after continual changing in graph. They describe symmetry of chemical structures with a number and then work for the improvement of QSAR and QSPR which both are employed to build a connection among the molecular structure and mathematical tools. These indices are useful to associate physiochemical properties of compounds (such as entropy, boiling and melting point, flammability, and many more).Topological indices are invariants of structures, so they are independent of pictorial representation [4]. Among three categories of molecular descriptors, vertex degree-based indices are considerably significant. Medicine industries are producing new and advanced medicines which are effective for mankind and ecology. Graph theory and molecular descriptors are playing a significant role in analysing the physiochemical properties of organic compounds.Hydrocarbon structure is an aromatic hydrocarbon and a unique structure composed of benzene through covalent bond. There are six sigma and six pi bonds in each benzene ring present in this compound. It is a nonpolar structure, and each benzene has a bond angle of 120°. It can be used in making of plastic, nylon, and dyes. It cannot be dissolved in water but in organic solvents. It has a sharp melting point because of the presence of benzene but does not have high boiling point. The structure is flammable and also show resonance.


### 1.1. Derivation of Degree-Based Topological Indices

#### 1.1.1. First General Zagreb Index

The 1st general Zagreb index was introduced by Li and Zhao and is given as [5](1)MαG∗=∑p∈VG∗Degpα.Classes of Zagreb Indices. We have two Zagreb groups of indices, first Zagreb index and second Zagreb index denoted by M1 and M2 [6–8]. They are proposed in late seventies by Gutman and Tranjistic.


#### 1.1.2. First Zagreb Index

The first Zagreb index can be written as [9, 10](2)M1G∗=∑pq∈EG∗Degp+Degq.


#### 1.1.3. Second Zagreb Index

The mathematical form is [9](3)M2G∗=∑pq∈EG∗Degp×Degq.Multiple and Polynomial Zagreb Indices. In 2012, advanced forms of Zagreb descriptors were suggested, with names 1st and 2nd multiple Zagreb descriptors given as PM1G∗ and PM2G∗ [8]. The polynomials are helpful to calculate Zageb index. 1st and 2nd Zagreb polynomial descriptors are denoted as M1G∗,j and M2G∗,j.


#### 1.1.4. First and Second Multiple Zagreb Indices

The 1st and 2nd multiple Zagreb indices are(4)PM1G∗=∏pq∈EG∗Degp+Degq,(5)PM2G∗=∏pq∈EG∗Degp×Degq.


#### 1.1.5. First and Second Polynomial Zagreb Indices

The 1st and 2nd polynomial Zagreb indices are(6)M1G∗,j=∑pq∈EG∗jDegp+Degq,(7)M2G∗,j=∑pq∈EG∗jDegp×Degq.


#### 1.1.6. Modified Zagreb Index

The modified form of Zagreb index was put forward in 2013 by G. H Shirdil, H. Rezapour, and A.M.Sayadi.(8)HMG∗=∑pq∈EG∗Degp+Degq2.


#### 1.1.7. Second Modified Zagreb Index

The 2nd modified Zagreb index is(9)M2G∗=∑pq∈EG∗1Degp×Degq.


#### 1.1.8. Reduced 2nd Zagreb Index

It was written by Furtula, and its formula is(10)RM2G∗=∑pq∈EG∗Degp−1×Degq−1.


#### 1.1.9. Atom Bond Connectivity Index

Ernesto Estrada and Torres defined the abovementioned index [8, 11]. It is helpful in modeling thermodynamic properties of hydrocarbons.(11)ABCG∗=∑pq∈EG∗Degp+Degq−2DegpDegq.


#### 1.1.10. Atom Bond Connectivity Index of 4th Order

Ghorbani and Ghazi suggested this index [7].(12)ABC4G∗=∑pq∈EG∗Sp+Sq−2SpSq.


#### 1.1.11. General Randić Connectivity Index

Millan Randić introduced molecular descriptors for the first time based on degree of vertices. Initially, it was coined as branching index [10] and familiar to find the branching of hydrocarbons. In 1998, Eddrös and Bollobás suggested the general form of this index by switching the factor −1/2 with αεI​R [12].(13)RαG∗=∑pq∈EG∗DegpDegqα.


#### 1.1.12. Randić Index

We may call this index as first-degree-based topological index [8].(14)RG∗=∑pq∈EG∗1DegpDegq.


#### 1.1.13. Reciprocal Randić Index (RRI)

Favaron, Mahéo, and Saclé invented a new index RRI [13].(15)RRG∗=∑pq∈EG∗DegpDegq.


#### 1.1.14. RRR Index

It is equivalent of RR index [13]. It can be written as(16)RRRG∗=∑pq∈EG∗Degp−1Degq−1.


#### 1.1.15. GA Index

Vukicevic̀ and Furtula proposed the GA index [4, 7, 8].(17)GAG∗=∑pq∈EG∗2DegpDegqDegp+Degq.


#### 1.1.16.GA5 Index

Grovac et al. suggested theGA5 index in 2011.(18)GA5G∗=∑pq∈EG∗2SpSqSp+Sq.


#### 1.1.17. Forgotten Index

Gutman and Futula suggested an index [14]. It is represented by FG∗.(19)FG∗=∑pq∈EG∗Degp2+Degq2.


#### 1.1.18. General Sum Connectivity Index

Zhou and Trinajstić proposed new index [8, 15]. It is defined as(20)χαG∗=∑pq∈EG∗Degp+Degqα,where αεI​R.


#### 1.1.19. SD Index

In 2010, D.vukicevic and Furtula proposed this useful index denoted by SD (G) [17–21].(21)SDG∗=∑pq∈EG∗Degp2+Degq2Degp×Degq.


#### 1.1.20. Harmonic Index

Siemion Fajtlowicz prepared a computer program that is helpful in automatic generation of conjectures [8] and also suggested a degree-based element; then, Zhang unwrapped this element and named harmonic index [22–25].(22)HG∗=∑pq∈EG∗2Degp+Degq.


## 1.1. Derivation of Degree-Based Topological Indices

### 1.1.1. First General Zagreb Index

The 1st general Zagreb index was introduced by Li and Zhao and is given as [5](1)MαG∗=∑p∈VG∗Degpα.Classes of Zagreb Indices. We have two Zagreb groups of indices, first Zagreb index and second Zagreb index denoted by M1 and M2 [6–8]. They are proposed in late seventies by Gutman and Tranjistic.


### 1.1.2. First Zagreb Index

The first Zagreb index can be written as [9, 10](2)M1G∗=∑pq∈EG∗Degp+Degq.


### 1.1.3. Second Zagreb Index

The mathematical form is [9](3)M2G∗=∑pq∈EG∗Degp×Degq.Multiple and Polynomial Zagreb Indices. In 2012, advanced forms of Zagreb descriptors were suggested, with names 1st and 2nd multiple Zagreb descriptors given as PM1G∗ and PM2G∗ [8]. The polynomials are helpful to calculate Zageb index. 1st and 2nd Zagreb polynomial descriptors are denoted as M1G∗,j and M2G∗,j.


### 1.1.4. First and Second Multiple Zagreb Indices

The 1st and 2nd multiple Zagreb indices are(4)PM1G∗=∏pq∈EG∗Degp+Degq,(5)PM2G∗=∏pq∈EG∗Degp×Degq.


### 1.1.5. First and Second Polynomial Zagreb Indices

The 1st and 2nd polynomial Zagreb indices are(6)M1G∗,j=∑pq∈EG∗jDegp+Degq,(7)M2G∗,j=∑pq∈EG∗jDegp×Degq.


### 1.1.6. Modified Zagreb Index

The modified form of Zagreb index was put forward in 2013 by G. H Shirdil, H. Rezapour, and A.M.Sayadi.(8)HMG∗=∑pq∈EG∗Degp+Degq2.


### 1.1.7. Second Modified Zagreb Index

The 2nd modified Zagreb index is(9)M2G∗=∑pq∈EG∗1Degp×Degq.


### 1.1.8. Reduced 2nd Zagreb Index

It was written by Furtula, and its formula is(10)RM2G∗=∑pq∈EG∗Degp−1×Degq−1.


### 1.1.9. Atom Bond Connectivity Index

Ernesto Estrada and Torres defined the abovementioned index [8, 11]. It is helpful in modeling thermodynamic properties of hydrocarbons.(11)ABCG∗=∑pq∈EG∗Degp+Degq−2DegpDegq.


### 1.1.10. Atom Bond Connectivity Index of 4th Order

Ghorbani and Ghazi suggested this index [7].(12)ABC4G∗=∑pq∈EG∗Sp+Sq−2SpSq.


### 1.1.11. General Randić Connectivity Index

Millan Randić introduced molecular descriptors for the first time based on degree of vertices. Initially, it was coined as branching index [10] and familiar to find the branching of hydrocarbons. In 1998, Eddrös and Bollobás suggested the general form of this index by switching the factor −1/2 with αεI​R [12].(13)RαG∗=∑pq∈EG∗DegpDegqα.


### 1.1.12. Randić Index

We may call this index as first-degree-based topological index [8].(14)RG∗=∑pq∈EG∗1DegpDegq.


### 1.1.13. Reciprocal Randić Index (RRI)

Favaron, Mahéo, and Saclé invented a new index RRI [13].(15)RRG∗=∑pq∈EG∗DegpDegq.


### 1.1.14. RRR Index

It is equivalent of RR index [13]. It can be written as(16)RRRG∗=∑pq∈EG∗Degp−1Degq−1.


### 1.1.15. GA Index

Vukicevic̀ and Furtula proposed the GA index [4, 7, 8].(17)GAG∗=∑pq∈EG∗2DegpDegqDegp+Degq.


### 1.1.16.GA5 Index

Grovac et al. suggested theGA5 index in 2011.(18)GA5G∗=∑pq∈EG∗2SpSqSp+Sq.


### 1.1.17. Forgotten Index

Gutman and Futula suggested an index [14]. It is represented by FG∗.(19)FG∗=∑pq∈EG∗Degp2+Degq2.


### 1.1.18. General Sum Connectivity Index

Zhou and Trinajstić proposed new index [8, 15]. It is defined as(20)χαG∗=∑pq∈EG∗Degp+Degqα,where αεI​R.


### 1.1.19. SD Index

In 2010, D.vukicevic and Furtula proposed this useful index denoted by SD (G) [17–21].(21)SDG∗=∑pq∈EG∗Degp2+Degq2Degp×Degq.


### 1.1.20. Harmonic Index

Siemion Fajtlowicz prepared a computer program that is helpful in automatic generation of conjectures [8] and also suggested a degree-based element; then, Zhang unwrapped this element and named harmonic index [22–25].(22)HG∗=∑pq∈EG∗2Degp+Degq.


## 1.1.1. First General Zagreb Index

The 1st general Zagreb index was introduced by Li and Zhao and is given as [5](1)MαG∗=∑p∈VG∗Degpα.Classes of Zagreb Indices. We have two Zagreb groups of indices, first Zagreb index and second Zagreb index denoted by M1 and M2 [6–8]. They are proposed in late seventies by Gutman and Tranjistic.


## 1.1.2. First Zagreb Index

The first Zagreb index can be written as [9, 10](2)M1G∗=∑pq∈EG∗Degp+Degq.


## 1.1.3. Second Zagreb Index

The mathematical form is [9](3)M2G∗=∑pq∈EG∗Degp×Degq.Multiple and Polynomial Zagreb Indices. In 2012, advanced forms of Zagreb descriptors were suggested, with names 1st and 2nd multiple Zagreb descriptors given as PM1G∗ and PM2G∗ [8]. The polynomials are helpful to calculate Zageb index. 1st and 2nd Zagreb polynomial descriptors are denoted as M1G∗,j and M2G∗,j.


## 1.1.4. First and Second Multiple Zagreb Indices

The 1st and 2nd multiple Zagreb indices are(4)PM1G∗=∏pq∈EG∗Degp+Degq,(5)PM2G∗=∏pq∈EG∗Degp×Degq.


## 1.1.5. First and Second Polynomial Zagreb Indices

The 1st and 2nd polynomial Zagreb indices are(6)M1G∗,j=∑pq∈EG∗jDegp+Degq,(7)M2G∗,j=∑pq∈EG∗jDegp×Degq.


## 1.1.6. Modified Zagreb Index

The modified form of Zagreb index was put forward in 2013 by G. H Shirdil, H. Rezapour, and A.M.Sayadi.(8)HMG∗=∑pq∈EG∗Degp+Degq2.


## 1.1.7. Second Modified Zagreb Index

The 2nd modified Zagreb index is(9)M2G∗=∑pq∈EG∗1Degp×Degq.


## 1.1.8. Reduced 2nd Zagreb Index

It was written by Furtula, and its formula is(10)RM2G∗=∑pq∈EG∗Degp−1×Degq−1.


## 1.1.9. Atom Bond Connectivity Index

Ernesto Estrada and Torres defined the abovementioned index [8, 11]. It is helpful in modeling thermodynamic properties of hydrocarbons.(11)ABCG∗=∑pq∈EG∗Degp+Degq−2DegpDegq.


## 1.1.10. Atom Bond Connectivity Index of 4th Order

Ghorbani and Ghazi suggested this index [7].(12)ABC4G∗=∑pq∈EG∗Sp+Sq−2SpSq.


## 1.1.11. General Randić Connectivity Index

Millan Randić introduced molecular descriptors for the first time based on degree of vertices. Initially, it was coined as branching index [10] and familiar to find the branching of hydrocarbons. In 1998, Eddrös and Bollobás suggested the general form of this index by switching the factor −1/2 with αεI​R [12].(13)RαG∗=∑pq∈EG∗DegpDegqα.


## 1.1.12. Randić Index

We may call this index as first-degree-based topological index [8].(14)RG∗=∑pq∈EG∗1DegpDegq.


## 1.1.13. Reciprocal Randić Index (RRI)

Favaron, Mahéo, and Saclé invented a new index RRI [13].(15)RRG∗=∑pq∈EG∗DegpDegq.


## 1.1.14. RRR Index

It is equivalent of RR index [13]. It can be written as(16)RRRG∗=∑pq∈EG∗Degp−1Degq−1.


## 1.1.15. GA Index

Vukicevic̀ and Furtula proposed the GA index [4, 7, 8].(17)GAG∗=∑pq∈EG∗2DegpDegqDegp+Degq.


## 1.1.16.GA5 Index

Grovac et al. suggested theGA5 index in 2011.(18)GA5G∗=∑pq∈EG∗2SpSqSp+Sq.


## 1.1.17. Forgotten Index

Gutman and Futula suggested an index [14]. It is represented by FG∗.(19)FG∗=∑pq∈EG∗Degp2+Degq2.


## 1.1.18. General Sum Connectivity Index

Zhou and Trinajstić proposed new index [8, 15]. It is defined as(20)χαG∗=∑pq∈EG∗Degp+Degqα,where αεI​R.


## 1.1.19. SD Index

In 2010, D.vukicevic and Furtula proposed this useful index denoted by SD (G) [17–21].(21)SDG∗=∑pq∈EG∗Degp2+Degq2Degp×Degq.


## 1.1.20. Harmonic Index

Siemion Fajtlowicz prepared a computer program that is helpful in automatic generation of conjectures [8] and also suggested a degree-based element; then, Zhang unwrapped this element and named harmonic index [22–25].(22)HG∗=∑pq∈EG∗2Degp+Degq.


## 2. Topological Indices of Hydrocarbon Structure

In this study, numbers of molecular descriptors of the hydrocarbon structure are computed (Figure1).Figure 1
Unit (2, 3) of hydrocarbon structure.


### 2.1. Description of Graph of Hydrocarbon Structure

Chemical properties of graph shown in Figure2 are given (Tables 1–4).Figure 2
Unit (1, 1) of hydrocarbon structure.Table 1
Chemical properties of graphG∗.
Chemical formulaC54H34Exact mass682.27Molecular weight682.85Elemental analysisC, 94.98; H, 5.02Table 2
Partition of graphG∗ on the basis of degrees.
Degree of vertexNumber of vertices230pq + 2p + 2q324pq − 2p − 2qSum66pq − q − pTable 3
Edge partition of graphG∗ with respect to starting and ending vertices of each edge.
Degp,Degq where p, qεE (G)Number of vertices(2,2)112pq + 4p + 4q(2,3)36pq − 4p − 4q(3,3)18pq − p − qSum66pq − p − qTable 4
Edge grouping of a graphG∗ of hydrocarbon structure with accordance of degree summation of adjoining end vertices of every edges.
Degree of verticesNumber of vertices(4, 4)2p(4, 5)4p + 4q(5, 5)12pq − 2p(5, 7)22pq − 2p(5, 8)2pq + 2p + 4q(6, 7)6pq − 2p − 4q(6, 8)6pq − 2p − 4q(7, 7)2pq − p − q(7, 8)8pq(7, 9)2pq(8, 8)2pq(8, 9)4pqSum66pq − p − qOur concerned graph is shown in Figure2, and it is denoted by G∗.Theorem 1.
G∗ is a graph of hydrocarbon structure, and its first general Zagreb index is given as follows:(23)MαG∗=48pq2α+18pq3α−q3α−p3αProof.
Consider graphG∗, i.e., shown in Figure 2. G∗ has 54pq points in which 30pq+2p+2q of degree 2 vertices and 24pq−2p−2q of degree 3.
By applying the definition ofMαG∗ (1),(24)MαG∗=∑p∈VG∗Degpα,we have the required results:(25)MαG∗=48pq2α+18pq3α−q3α−p3αTheorem 2.
First Zagreb index of graphG∗ is given as follows:(26)M1G∗=12pq4+4p4+4q4+36pq5−4p5−4q5+18pq6−p6−q6.Proof.
EG∗ of G∗ is divided into 3 groups.E1G∗ holds 12pq+4p+4q arcs pq, here Degp=Degq=2E2G∗ has 36pq−4p−4q arcs pq, here Degp=2, Degq=3E3G∗ contains 18pq−q−p arcs pq, here Degp=3, Degq=3ConsiderE1G∗=e2,2E2G∗=e2,3E2G∗=e3,3
From equation (6), we get(27)M1G∗=∑pq∈EG∗Degp+Degq,M1G∗=∑pq∈E1G∗Degp+Degq+∑pq∈E2G∗Degp+Degq+∑pq∈E3G∗Degp+Degq=E1G∗4+E2G∗5+E3G∗6=12pq+4p+4q4+36pq−4p−4q5+18pq−q−p6=12pq4+4p4+4q4+36pq5−4p5−4q5+18pq6−q6−p6.Theorem 3.
First and second polynomial and multiple Zagreb indices of (hydrocarbon structure)G∗ are given as follows:(1)
PM1G∗=412pq+4p+4q×536pq−4p−4q×618pq−q−p;(2)
PM2G∗=412pq+4p+4q×636pq−4p−4q×918pq−q−p;(3)
M1G∗,j=12pq+4p+4qj4+36pq−4p−4qj5+18pq−p−qj6;(4)
M2G∗,j=12pq+4p+4qj4+36pq−4p−4qj6+18pq−p−qj9;Proof.
EG∗ is grouped in 3 edge partitions depending on end vertices degrees. E1G∗ contains 12pq+4p+4q edges, where Degp=Degq=2. E2G∗ has 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q lines pq, where Degp=3 and Degq=3. Consider E1G∗=e2,2, E2G∗=e2,3, and E2G∗=e3,3.
By utilizing the definition ofPM1G∗,(28)PM1G∗=∏pq∈EG∗Degp+Degq,PM1G∗=∏pq∈E1G∗Degp+Degq×∏pq∈E2G∗Degp+Degq×∏pq∈E3G∗Degp+Degq=4E1G∗×5E2G∗×6E3G∗=412pq+4p+4q×536pq−4p−4q×618pq−q−p.
Now,(29)PM2G∗=∏pq∈EG∗Degp×Degq,PM2G∗=∏pq∈E1G∗Degp×Degq×∏pq∈E2G∗Degp×Degq×∏pq∈E3G∗Degp×Degq=4E1G∗×5E2G∗×6E3G∗=412pq+4p+4q×536pq−4p−4q×618pq−p−q.
By utilizingM1G∗,j from (6),(30)M1G∗,j=∑pq∈EG∗jDegp+Degq,M1G∗,j=∑pq∈E1G∗jDegp+Degq+∑pq∈E2G∗jDegp+Degq+∑pq∈E3G∗jDegp+Degq=∑pq∈E1G∗j4+∑pq∈E1G∗j5+∑pq∈E1G∗j6=E1G∗j4+E2G∗j5+E3G∗j6=12pq+4p+4qj4+36pq−4p−4qj5+18pq−q−pj6.
From equation (7), we have(31)M2G∗,j=∑pq∈EG∗jDegp+Degq,M2G∗,j=∑pq∈E1G∗jDegp+Degq+∑pq∈E2G∗jDegp+Degq+∑pq∈E3G∗jDegp+Degq=∑pq∈E1G∗j4+∑pq∈E1G∗j6+∑pq∈E1G∗j9=E1G∗j4+E2G∗j6+E3G∗j9=12pq+4p+4qj4+36pq−4p−4qj6+18pq−q−pj9.
This completes the proof.Theorem 4.
Harmonic index, second Zagreb index, and reduced second Zagreb index ofG∗ are as follows:(1)
Hyper-Zagreb index of graphG∗ is(32)HMG∗=12pq16+4p16+4q16+36pq25−4p25−4q25+18pq36−q36−p36.(2)
Second Zagreb index is(33)M2G∗=11pq+29p+29q.(3)
Reduced2nd Zagreb index is(34)RM2G∗=−20p−24q+144pq.Proof.
EG∗ is grouped in 3 partitions. E1G∗ holds 12pq+4p+4q edges, where Degp=Degq=2. E2G∗ supports 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ keeps 18pq−q−p edges, where Degp=3 and Degq=3. Consider E1G∗=e2,2, E2G∗=e2,3, and E2G∗=e3,3.
From (8), we define HMG∗ as(35)HMG∗=∑pq∈EG∗Degp+Degq2,HMG∗=∑pq∈E1G∗Degp+Degq2+∑pq∈E2G∗Degp+Degq2+∑pq∈E3G∗Degp+Degq2=16E1G∗+25E2G∗+36E3G∗=1612pq+4p+4q+2536pq−4p−4q+3618pq−q−p=1740pq−72p−72q.
By using the definition ofM2G∗,(36)M2G∗=∑pq∈EG∗1Degp+Degq,M2G∗=E1G∗14+E2G∗16+E3G∗19=12pq+4p+4q4+36pq−4p−4q6+18pq−q−p9=11pq+29p+29q.
By substituting the values in equation (10),(37)RM2G∗=∑pq∈EG∗Degp−1×Degq−1.
With the help of (3) and (4), we have(38)RM2G∗=E1G∗11+E2G∗12+E3G∗22=12pq+4p+4q+36pq−4p−4q2+18pq−q−p4=12pq+4p+4q+36pq2−4p2−4q2+18pq4−q4−p4.Theorem 5.
ABC index of graph hydrocarbon structure is given as follows:(39)ABCG∗=48pq122+18pq23−q23−p23.Proof.
G∗ encounters 66pq−p−q number of edges and 54pq vertices. Vertex count of degree 2 is 30pq+2p+2q and of degree 3 is 24pq−2p−2q. The cardinality arc group E of G∗ is 66pq−p−q. EG∗ grouped into 3 disjoint arc groups, i.e., EG∗=E1G∗∪E2G∗∪E3G∗. E1G∗ has 12mn+4n+4m edges pq, where Degp=Degq=2. E2G∗ supports 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q arcs pq, where Degp=Degq=3.
We useABCG∗ in (11) as(40)ABCG∗=∑pq∈EG∗Degp+Degq−2DegpDegq,ABCG∗=E1G∗122+E2G∗122+E3G∗23=12pq+4p+4q122+3pq−4p−4q122+18pq−q−p23=12pq12+4p12+4q122+36pq12−4q12−4p122+18pq23−q23−p23.Theorem 6.
(1)
ABC4G∗ of G∗ is(41)ABC4G∗=2p146+4p11035+4q11035+12pq252−2p252+22pq1714−2p1714+2pq120110+2p120110+4q120110+6q142462−2p142462−4q142462+6pq12−2p12−4q12+2pq273−p273−q273+8pq128182+2pq132+2pq1814+4pq11230.(2)
GA5G∗ is(42)GA5G∗=2p+4n495+4q495+14pq1−3p1+22pq1635−2p1635+2pq41310+2p41310+4q41310+6pq21342−2p21342−4q21342+6pq473−2p473−4q473−q1+8pq41514+pq347+pq2+4pq12172.Proof.
The graphG∗ has 66pq − q − p number of edges. EG∗ can be distributed into twelve disunite groups of edges.
EiG∗, i=4,5,6…,15. EG∗=∪i=415EiG∗.
E4G∗ has 2n lines pq, where Sp=Sq=4. E5G∗ supports 4p+4q lines pq, where Sp=4 and Sq=5. E6G∗ contains 12pq−2p edges, where Sp=Sq=5. E7G∗ contains 22pq−2p edges, where Sp=5 and Sq=7. E8G∗ keeps 2pq+2p+4q edges, where Sp=5 and Sq=8. E9G∗ contains 6pq−2p−4q edges, where Sp=6 and Sq=7. E10G∗ contains 6pq−2p−4q edges, where Sp=6 and Sq=8. E11G∗ contains 2pq−p−q edges, where Sp=Sq=7. E12G∗ holds 8pq edges, where Sp=7 and Sq=8. E13G∗ holds 2pq edges, where Sp=7 and Sq=9. The edge set E14G∗ keep 2mn edges, where Sp=Sq=8. E15G∗ holds 4mn edges, here pq, where Sp=8 and Sq=9.
The index is defined in equation (12):(43)ABC4G∗=∑pq∈EG∗Sp+Sq−2SpSq,(44)ABC4G∗=4+4−24×4E4G∗+4+5−24×5E5G∗+5+5−25×5E6G∗+5+7−25×7E7G∗+5+8−25×8E8G∗+6+7−26×7E9G∗+6+8−26×8E10G∗+7+7−27×7E11G∗+7+8−27×8E12G∗+7+9−27×9E13G∗+8+8−28×8E14G∗+8+9−28×9E15G∗=616E4G∗+720E5G∗+825E6G∗+1035E7G∗+1140E8G∗+1142E9G∗+1248E10G∗+1249E11G∗+1356E12G∗+1463E13G∗+1464E14G∗+1572E15G∗.
After substituting the valuesEG∗=∪i=513EiG∗, we get(45)=6162p+7204p+4q+82512pq−2q+103522pq−2p+11402pq+2p+4q+11426pq−2p−4q+12486pq−2p−4q+12492pq−p−q+13568mn+14632pq+15724pq.
After simplification, we get(46)=126p+110354p+4q+25212pq−2p+171422pq−2p+1201102pq+2p+4q+1424626pq−2p−4q+3pq−p−2q+2732pq−q−p+27182pq+232pq+1330pq.
By utilizing the definition ofGA5G∗ from equation (18),(47)GA5G∗=∑pq∈EG∗2SpSqSp+Sq,(48)GA5G∗=24×44+4E4G∗+24×54+5E5G∗+25×55+5E6G∗+25×75+7E7G∗+25×85+8E8G∗+26×76+7E9G∗+26×86+8E10G∗+27×77+7E11G∗+27×87+8E12G∗+27×97+9E13G∗+28×88+8E14G∗+28×98+9E15G∗.
After substituting the valuesEG∗=∪i=513EiG∗, we get(49)=21682p+22094p+4q+2251012pq−2p+2351222pq−2p+240132pq+2p+4q+242136pq−2p−4q+248146pq−2p−4q+249142pq−q−p+256158pq+263162pq++264162pq+272174pq.
After simplification,(50)=−p+4954p+4q+16pq+163522pq−2p+413102pq+2p+4q+213426pq−2p−4q+4736pq−2p−4q−q+321514pq+347pq+48172pq.Theorem 7.
Consider the following:(1)
The general Randić index of graphG∗ is given as follows:(51)RαG∗=12pq4α+4p4α+4q4α+36pq6α−4p6α−4q6α+18pq9α−q9α−p9α.(2)
Randić index of graphG∗ is(52)RG∗=12pq+53p+53q+16636pq−4p−4q.(3)
Reduced reciprocal Randić index of graphG∗ is(53)RRRG∗=48pq+2p+2q+236pq−4p−4q.(4)
Reciprocal Randić index of graphG∗ is(54)RRG∗=78pq+5p+5q+636pq−4p−4q.Proof.
G∗ encounters 66pq−p−q lines and 54pq vertices. Vertices of degree 2 are 30pq+2p+2q and of degree 3 are 24pq−2p−2q. The cardinality of E of G∗ is 66pq−p−q. EG∗ is divided into 3 dissociate edge groups that rely on the degrees of the end points, i.e., EG∗=E1G∗∪E2G∗∪E3G∗. E1G∗ has 12pq+4p+4q edges pq, where Degp=Degq=2. E2G∗ has 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q arcs pq, where Degp=Degq=3.
We use general Randic index in (13) as(55)RαG∗=∑pq∈EG∗DegpDegqα.
Now, we have(56)RαG∗=∑pq∈E1G∗DegpDegqα+∑pq∈E2G∗DegpDegqα+∑xy∈E3G∗DegpDegqα=4E1G∗+6E2G∗+9E3G∗=412pq+4p+4q+636pq−4q−4p+918pq−p−q.
After simplification, we get(57)=426pq−17p−17q.
By the use of Randić index (14),(58)RG∗=∑pq∈EG∗1DegpDegq,RG∗=∑pq∈E1G∗1DegpDegq+∑pq∈E2G∗1DegpDegq+∑pq∈E3G∗1DegpDegq=12E1G∗+16E2G∗+13E3G∗=1212pq+4p+4q+1636pq−4q−4p+1318pq−p−q.
After simplification,(59)RG∗=12pq+53n+53q+16636pq−4p−4q.
Definition ofRRRG∗ index from equation (16) is(60)RRRG∗=∑pq∈EG∗Degp−1Degq−1,RRRG∗=∑pq∈E1G∗Degp−1Degq−1+∑pq∈E2G∗Degp−1Degq−1+∑pq∈E3G∗Degp−1Degq−1=1E1G∗+2E2G∗+2E3G∗=12pq+4p+4q+236pq−4q−4p+218pq−p−q=48pq+2p+2q+236pq−4q−4p.
Now, by utilizing the definition of reduced Randić index from equation (15),(61)RRG∗=∑pq∈EG∗DegpDegq,=∑pq∈E1G∗DegpDegq+∑pq∈E2G∗DegpDegq+∑pq∈E3G∗DegpDegq=2E1G∗+6E2G∗+3E3G∗=212pq+4p+4q+636pq−4q−4p+318pq−p−q.Theorem 8.
We have graphG∗ and its different indices are explained here.(1)
GA index is as follows:(62)GAG∗=30pq+3p+3q+36pq256−4p256−4q256.(2)
Sum connectivity index is given as follows:(63)XαG∗=12pq4α+4p4α+4q4α+36pq5α−4p5α−4q5α+18pq6α−q6α−p6α.(3)
Forgotten index ofG∗ is(64)FG∗=12pq8+4p8+4q8+36pq13−4p13−4q13+18pq18−q18−p18.Proof.
G∗ encounters 66pq−p−q edges and 54pq vertices. Vertex count of degree 2 are 30pq+2p+2q and of degree 3 are 24pq−2p−2q. The cardinality line group E of G∗ is 66pq−p−q. EG∗ is classified into three disjoint edge groups, i.e., EG∗=E1G∗∪E2G∗∪E3G∗. E1G∗ has 12pq+4p+4q edges pq, where Degp=Degq=2. E2G∗ has 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q edges pq, where Degp=Degq=3.
We use geometric arithmetic index in (17) as(65)GAG∗=∑pq∈EG∗2DegpDegqDegp+Degq,GAG∗=∑pq∈E1G∗2DegpDegqDegp+Degq+∑pq∈E2G∗2DegpDegqDegp+Degq+∑pq∈E3G∗2DegpDegqDegp+Degq=1E1G∗+256E2G∗+1E3G∗=12pq+4p+4q+636pq−4q−4p+18pq−p−q.
After simplification, we obtain(66)=30pq+3p+3q+636pq−4q−4p.
From equation (20), we get(67)χαG∗=∑pq∈EG∗Degp+Degqα,χαG∗=∑pq∈E1G∗Degp+Degqα+∑pq∈E2G∗Degp+Degqα+∑pq∈E3G∗Degp+Degqα=4αE1G∗+5αE2G∗+6αE3G∗=4α12pq+4p+4q+5α36pq−4q−4p+6α18pq−p−q.
We use forgotten index in (19) as(68)FG∗=∑pq∈EG∗Degp2+Degq2,FG∗=∑pq∈E1G∗Degp2+Degq2+∑pq∈E2G∗Degp2+Degq2+∑pq∈E3G∗Degp2+Degq2=8E1G∗+13E2G∗+18E3G∗=812pq+4p+4q+1336pq−4q−4p+1818pq−p−q=888pq−38p−38q.Theorem 9.
LetG∗ be graph:(1)
Symmetric division index is(69)SDG∗=138pq−83p−83q.(2)
Harmonic index is(70)HG∗=1325pq+115p+115q.Proof.
G∗ encounters 66pq−p−q edge and 54pq points. Vertex counts of degree 2 are 30pq+2p+2q and of degree 3 are 24pq−2p−2q. The edge set EG∗ splits into three distinct line groups.
EG∗=E1G∗∪E2G∗∪E3G∗E1G∗ has 12pq+4p+4q lines pq, where Degp=Degq=2. E2G∗ has 36pq−4p−4q lines pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q lines pq, where Degp=Degq=3.
From equation (21), we obtain(71)SDG∗=∑pq∈EG∗Degp2+Degq2Degp×DegqSDG∗=∑pq∈E1G∗Degp2+Degq2Degp×Degq+∑pq∈E2G∗Degp2+Degq2Degp×Degq+∑pq∈E3G∗Degp2+Degq2Degp×Degq=2E1G∗+135E2G∗+2E3G∗=212pq+4p+4q+13536pq−4q−4p+218pq−p−q.
From equation (22),(72)HG∗=∑pq∈EG∗2Degp×DegqHG∗=∑pq∈E1G∗2Degp×Degq+∑pq∈E2G∗2Degp×Degq+∑pq∈E3G∗2Degp×Degq=24E1G∗+25E2G∗+26E3G∗=1212pq+4p+4q+2536pq−4q−4p+1318pq−p−q=1325pq+1915p+1915q.


## 2.1. Description of Graph of Hydrocarbon Structure

Chemical properties of graph shown in Figure2 are given (Tables 1–4).Figure 2
Unit (1, 1) of hydrocarbon structure.Table 1
Chemical properties of graphG∗.
Chemical formulaC54H34Exact mass682.27Molecular weight682.85Elemental analysisC, 94.98; H, 5.02Table 2
Partition of graphG∗ on the basis of degrees.
Degree of vertexNumber of vertices230pq + 2p + 2q324pq − 2p − 2qSum66pq − q − pTable 3
Edge partition of graphG∗ with respect to starting and ending vertices of each edge.
Degp,Degq where p, qεE (G)Number of vertices(2,2)112pq + 4p + 4q(2,3)36pq − 4p − 4q(3,3)18pq − p − qSum66pq − p − qTable 4
Edge grouping of a graphG∗ of hydrocarbon structure with accordance of degree summation of adjoining end vertices of every edges.
Degree of verticesNumber of vertices(4, 4)2p(4, 5)4p + 4q(5, 5)12pq − 2p(5, 7)22pq − 2p(5, 8)2pq + 2p + 4q(6, 7)6pq − 2p − 4q(6, 8)6pq − 2p − 4q(7, 7)2pq − p − q(7, 8)8pq(7, 9)2pq(8, 8)2pq(8, 9)4pqSum66pq − p − qOur concerned graph is shown in Figure2, and it is denoted by G∗.Theorem 1.
G∗ is a graph of hydrocarbon structure, and its first general Zagreb index is given as follows:(23)MαG∗=48pq2α+18pq3α−q3α−p3αProof.
Consider graphG∗, i.e., shown in Figure 2. G∗ has 54pq points in which 30pq+2p+2q of degree 2 vertices and 24pq−2p−2q of degree 3.
By applying the definition ofMαG∗ (1),(24)MαG∗=∑p∈VG∗Degpα,we have the required results:(25)MαG∗=48pq2α+18pq3α−q3α−p3αTheorem 2.
First Zagreb index of graphG∗ is given as follows:(26)M1G∗=12pq4+4p4+4q4+36pq5−4p5−4q5+18pq6−p6−q6.Proof.
EG∗ of G∗ is divided into 3 groups.E1G∗ holds 12pq+4p+4q arcs pq, here Degp=Degq=2E2G∗ has 36pq−4p−4q arcs pq, here Degp=2, Degq=3E3G∗ contains 18pq−q−p arcs pq, here Degp=3, Degq=3ConsiderE1G∗=e2,2E2G∗=e2,3E2G∗=e3,3
From equation (6), we get(27)M1G∗=∑pq∈EG∗Degp+Degq,M1G∗=∑pq∈E1G∗Degp+Degq+∑pq∈E2G∗Degp+Degq+∑pq∈E3G∗Degp+Degq=E1G∗4+E2G∗5+E3G∗6=12pq+4p+4q4+36pq−4p−4q5+18pq−q−p6=12pq4+4p4+4q4+36pq5−4p5−4q5+18pq6−q6−p6.Theorem 3.
First and second polynomial and multiple Zagreb indices of (hydrocarbon structure)G∗ are given as follows:(1)
PM1G∗=412pq+4p+4q×536pq−4p−4q×618pq−q−p;(2)
PM2G∗=412pq+4p+4q×636pq−4p−4q×918pq−q−p;(3)
M1G∗,j=12pq+4p+4qj4+36pq−4p−4qj5+18pq−p−qj6;(4)
M2G∗,j=12pq+4p+4qj4+36pq−4p−4qj6+18pq−p−qj9;Proof.
EG∗ is grouped in 3 edge partitions depending on end vertices degrees. E1G∗ contains 12pq+4p+4q edges, where Degp=Degq=2. E2G∗ has 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q lines pq, where Degp=3 and Degq=3. Consider E1G∗=e2,2, E2G∗=e2,3, and E2G∗=e3,3.
By utilizing the definition ofPM1G∗,(28)PM1G∗=∏pq∈EG∗Degp+Degq,PM1G∗=∏pq∈E1G∗Degp+Degq×∏pq∈E2G∗Degp+Degq×∏pq∈E3G∗Degp+Degq=4E1G∗×5E2G∗×6E3G∗=412pq+4p+4q×536pq−4p−4q×618pq−q−p.
Now,(29)PM2G∗=∏pq∈EG∗Degp×Degq,PM2G∗=∏pq∈E1G∗Degp×Degq×∏pq∈E2G∗Degp×Degq×∏pq∈E3G∗Degp×Degq=4E1G∗×5E2G∗×6E3G∗=412pq+4p+4q×536pq−4p−4q×618pq−p−q.
By utilizingM1G∗,j from (6),(30)M1G∗,j=∑pq∈EG∗jDegp+Degq,M1G∗,j=∑pq∈E1G∗jDegp+Degq+∑pq∈E2G∗jDegp+Degq+∑pq∈E3G∗jDegp+Degq=∑pq∈E1G∗j4+∑pq∈E1G∗j5+∑pq∈E1G∗j6=E1G∗j4+E2G∗j5+E3G∗j6=12pq+4p+4qj4+36pq−4p−4qj5+18pq−q−pj6.
From equation (7), we have(31)M2G∗,j=∑pq∈EG∗jDegp+Degq,M2G∗,j=∑pq∈E1G∗jDegp+Degq+∑pq∈E2G∗jDegp+Degq+∑pq∈E3G∗jDegp+Degq=∑pq∈E1G∗j4+∑pq∈E1G∗j6+∑pq∈E1G∗j9=E1G∗j4+E2G∗j6+E3G∗j9=12pq+4p+4qj4+36pq−4p−4qj6+18pq−q−pj9.
This completes the proof.Theorem 4.
Harmonic index, second Zagreb index, and reduced second Zagreb index ofG∗ are as follows:(1)
Hyper-Zagreb index of graphG∗ is(32)HMG∗=12pq16+4p16+4q16+36pq25−4p25−4q25+18pq36−q36−p36.(2)
Second Zagreb index is(33)M2G∗=11pq+29p+29q.(3)
Reduced2nd Zagreb index is(34)RM2G∗=−20p−24q+144pq.Proof.
EG∗ is grouped in 3 partitions. E1G∗ holds 12pq+4p+4q edges, where Degp=Degq=2. E2G∗ supports 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ keeps 18pq−q−p edges, where Degp=3 and Degq=3. Consider E1G∗=e2,2, E2G∗=e2,3, and E2G∗=e3,3.
From (8), we define HMG∗ as(35)HMG∗=∑pq∈EG∗Degp+Degq2,HMG∗=∑pq∈E1G∗Degp+Degq2+∑pq∈E2G∗Degp+Degq2+∑pq∈E3G∗Degp+Degq2=16E1G∗+25E2G∗+36E3G∗=1612pq+4p+4q+2536pq−4p−4q+3618pq−q−p=1740pq−72p−72q.
By using the definition ofM2G∗,(36)M2G∗=∑pq∈EG∗1Degp+Degq,M2G∗=E1G∗14+E2G∗16+E3G∗19=12pq+4p+4q4+36pq−4p−4q6+18pq−q−p9=11pq+29p+29q.
By substituting the values in equation (10),(37)RM2G∗=∑pq∈EG∗Degp−1×Degq−1.
With the help of (3) and (4), we have(38)RM2G∗=E1G∗11+E2G∗12+E3G∗22=12pq+4p+4q+36pq−4p−4q2+18pq−q−p4=12pq+4p+4q+36pq2−4p2−4q2+18pq4−q4−p4.Theorem 5.
ABC index of graph hydrocarbon structure is given as follows:(39)ABCG∗=48pq122+18pq23−q23−p23.Proof.
G∗ encounters 66pq−p−q number of edges and 54pq vertices. Vertex count of degree 2 is 30pq+2p+2q and of degree 3 is 24pq−2p−2q. The cardinality arc group E of G∗ is 66pq−p−q. EG∗ grouped into 3 disjoint arc groups, i.e., EG∗=E1G∗∪E2G∗∪E3G∗. E1G∗ has 12mn+4n+4m edges pq, where Degp=Degq=2. E2G∗ supports 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q arcs pq, where Degp=Degq=3.
We useABCG∗ in (11) as(40)ABCG∗=∑pq∈EG∗Degp+Degq−2DegpDegq,ABCG∗=E1G∗122+E2G∗122+E3G∗23=12pq+4p+4q122+3pq−4p−4q122+18pq−q−p23=12pq12+4p12+4q122+36pq12−4q12−4p122+18pq23−q23−p23.Theorem 6.
(1)
ABC4G∗ of G∗ is(41)ABC4G∗=2p146+4p11035+4q11035+12pq252−2p252+22pq1714−2p1714+2pq120110+2p120110+4q120110+6q142462−2p142462−4q142462+6pq12−2p12−4q12+2pq273−p273−q273+8pq128182+2pq132+2pq1814+4pq11230.(2)
GA5G∗ is(42)GA5G∗=2p+4n495+4q495+14pq1−3p1+22pq1635−2p1635+2pq41310+2p41310+4q41310+6pq21342−2p21342−4q21342+6pq473−2p473−4q473−q1+8pq41514+pq347+pq2+4pq12172.Proof.
The graphG∗ has 66pq − q − p number of edges. EG∗ can be distributed into twelve disunite groups of edges.
EiG∗, i=4,5,6…,15. EG∗=∪i=415EiG∗.
E4G∗ has 2n lines pq, where Sp=Sq=4. E5G∗ supports 4p+4q lines pq, where Sp=4 and Sq=5. E6G∗ contains 12pq−2p edges, where Sp=Sq=5. E7G∗ contains 22pq−2p edges, where Sp=5 and Sq=7. E8G∗ keeps 2pq+2p+4q edges, where Sp=5 and Sq=8. E9G∗ contains 6pq−2p−4q edges, where Sp=6 and Sq=7. E10G∗ contains 6pq−2p−4q edges, where Sp=6 and Sq=8. E11G∗ contains 2pq−p−q edges, where Sp=Sq=7. E12G∗ holds 8pq edges, where Sp=7 and Sq=8. E13G∗ holds 2pq edges, where Sp=7 and Sq=9. The edge set E14G∗ keep 2mn edges, where Sp=Sq=8. E15G∗ holds 4mn edges, here pq, where Sp=8 and Sq=9.
The index is defined in equation (12):(43)ABC4G∗=∑pq∈EG∗Sp+Sq−2SpSq,(44)ABC4G∗=4+4−24×4E4G∗+4+5−24×5E5G∗+5+5−25×5E6G∗+5+7−25×7E7G∗+5+8−25×8E8G∗+6+7−26×7E9G∗+6+8−26×8E10G∗+7+7−27×7E11G∗+7+8−27×8E12G∗+7+9−27×9E13G∗+8+8−28×8E14G∗+8+9−28×9E15G∗=616E4G∗+720E5G∗+825E6G∗+1035E7G∗+1140E8G∗+1142E9G∗+1248E10G∗+1249E11G∗+1356E12G∗+1463E13G∗+1464E14G∗+1572E15G∗.
After substituting the valuesEG∗=∪i=513EiG∗, we get(45)=6162p+7204p+4q+82512pq−2q+103522pq−2p+11402pq+2p+4q+11426pq−2p−4q+12486pq−2p−4q+12492pq−p−q+13568mn+14632pq+15724pq.
After simplification, we get(46)=126p+110354p+4q+25212pq−2p+171422pq−2p+1201102pq+2p+4q+1424626pq−2p−4q+3pq−p−2q+2732pq−q−p+27182pq+232pq+1330pq.
By utilizing the definition ofGA5G∗ from equation (18),(47)GA5G∗=∑pq∈EG∗2SpSqSp+Sq,(48)GA5G∗=24×44+4E4G∗+24×54+5E5G∗+25×55+5E6G∗+25×75+7E7G∗+25×85+8E8G∗+26×76+7E9G∗+26×86+8E10G∗+27×77+7E11G∗+27×87+8E12G∗+27×97+9E13G∗+28×88+8E14G∗+28×98+9E15G∗.
After substituting the valuesEG∗=∪i=513EiG∗, we get(49)=21682p+22094p+4q+2251012pq−2p+2351222pq−2p+240132pq+2p+4q+242136pq−2p−4q+248146pq−2p−4q+249142pq−q−p+256158pq+263162pq++264162pq+272174pq.
After simplification,(50)=−p+4954p+4q+16pq+163522pq−2p+413102pq+2p+4q+213426pq−2p−4q+4736pq−2p−4q−q+321514pq+347pq+48172pq.Theorem 7.
Consider the following:(1)
The general Randić index of graphG∗ is given as follows:(51)RαG∗=12pq4α+4p4α+4q4α+36pq6α−4p6α−4q6α+18pq9α−q9α−p9α.(2)
Randić index of graphG∗ is(52)RG∗=12pq+53p+53q+16636pq−4p−4q.(3)
Reduced reciprocal Randić index of graphG∗ is(53)RRRG∗=48pq+2p+2q+236pq−4p−4q.(4)
Reciprocal Randić index of graphG∗ is(54)RRG∗=78pq+5p+5q+636pq−4p−4q.Proof.
G∗ encounters 66pq−p−q lines and 54pq vertices. Vertices of degree 2 are 30pq+2p+2q and of degree 3 are 24pq−2p−2q. The cardinality of E of G∗ is 66pq−p−q. EG∗ is divided into 3 dissociate edge groups that rely on the degrees of the end points, i.e., EG∗=E1G∗∪E2G∗∪E3G∗. E1G∗ has 12pq+4p+4q edges pq, where Degp=Degq=2. E2G∗ has 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q arcs pq, where Degp=Degq=3.
We use general Randic index in (13) as(55)RαG∗=∑pq∈EG∗DegpDegqα.
Now, we have(56)RαG∗=∑pq∈E1G∗DegpDegqα+∑pq∈E2G∗DegpDegqα+∑xy∈E3G∗DegpDegqα=4E1G∗+6E2G∗+9E3G∗=412pq+4p+4q+636pq−4q−4p+918pq−p−q.
After simplification, we get(57)=426pq−17p−17q.
By the use of Randić index (14),(58)RG∗=∑pq∈EG∗1DegpDegq,RG∗=∑pq∈E1G∗1DegpDegq+∑pq∈E2G∗1DegpDegq+∑pq∈E3G∗1DegpDegq=12E1G∗+16E2G∗+13E3G∗=1212pq+4p+4q+1636pq−4q−4p+1318pq−p−q.
After simplification,(59)RG∗=12pq+53n+53q+16636pq−4p−4q.
Definition ofRRRG∗ index from equation (16) is(60)RRRG∗=∑pq∈EG∗Degp−1Degq−1,RRRG∗=∑pq∈E1G∗Degp−1Degq−1+∑pq∈E2G∗Degp−1Degq−1+∑pq∈E3G∗Degp−1Degq−1=1E1G∗+2E2G∗+2E3G∗=12pq+4p+4q+236pq−4q−4p+218pq−p−q=48pq+2p+2q+236pq−4q−4p.
Now, by utilizing the definition of reduced Randić index from equation (15),(61)RRG∗=∑pq∈EG∗DegpDegq,=∑pq∈E1G∗DegpDegq+∑pq∈E2G∗DegpDegq+∑pq∈E3G∗DegpDegq=2E1G∗+6E2G∗+3E3G∗=212pq+4p+4q+636pq−4q−4p+318pq−p−q.Theorem 8.
We have graphG∗ and its different indices are explained here.(1)
GA index is as follows:(62)GAG∗=30pq+3p+3q+36pq256−4p256−4q256.(2)
Sum connectivity index is given as follows:(63)XαG∗=12pq4α+4p4α+4q4α+36pq5α−4p5α−4q5α+18pq6α−q6α−p6α.(3)
Forgotten index ofG∗ is(64)FG∗=12pq8+4p8+4q8+36pq13−4p13−4q13+18pq18−q18−p18.Proof.
G∗ encounters 66pq−p−q edges and 54pq vertices. Vertex count of degree 2 are 30pq+2p+2q and of degree 3 are 24pq−2p−2q. The cardinality line group E of G∗ is 66pq−p−q. EG∗ is classified into three disjoint edge groups, i.e., EG∗=E1G∗∪E2G∗∪E3G∗. E1G∗ has 12pq+4p+4q edges pq, where Degp=Degq=2. E2G∗ has 36pq−4p−4q edges pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q edges pq, where Degp=Degq=3.
We use geometric arithmetic index in (17) as(65)GAG∗=∑pq∈EG∗2DegpDegqDegp+Degq,GAG∗=∑pq∈E1G∗2DegpDegqDegp+Degq+∑pq∈E2G∗2DegpDegqDegp+Degq+∑pq∈E3G∗2DegpDegqDegp+Degq=1E1G∗+256E2G∗+1E3G∗=12pq+4p+4q+636pq−4q−4p+18pq−p−q.
After simplification, we obtain(66)=30pq+3p+3q+636pq−4q−4p.
From equation (20), we get(67)χαG∗=∑pq∈EG∗Degp+Degqα,χαG∗=∑pq∈E1G∗Degp+Degqα+∑pq∈E2G∗Degp+Degqα+∑pq∈E3G∗Degp+Degqα=4αE1G∗+5αE2G∗+6αE3G∗=4α12pq+4p+4q+5α36pq−4q−4p+6α18pq−p−q.
We use forgotten index in (19) as(68)FG∗=∑pq∈EG∗Degp2+Degq2,FG∗=∑pq∈E1G∗Degp2+Degq2+∑pq∈E2G∗Degp2+Degq2+∑pq∈E3G∗Degp2+Degq2=8E1G∗+13E2G∗+18E3G∗=812pq+4p+4q+1336pq−4q−4p+1818pq−p−q=888pq−38p−38q.Theorem 9.
LetG∗ be graph:(1)
Symmetric division index is(69)SDG∗=138pq−83p−83q.(2)
Harmonic index is(70)HG∗=1325pq+115p+115q.Proof.
G∗ encounters 66pq−p−q edge and 54pq points. Vertex counts of degree 2 are 30pq+2p+2q and of degree 3 are 24pq−2p−2q. The edge set EG∗ splits into three distinct line groups.
EG∗=E1G∗∪E2G∗∪E3G∗E1G∗ has 12pq+4p+4q lines pq, where Degp=Degq=2. E2G∗ has 36pq−4p−4q lines pq, where Degp=2 and Degq=3. E3G∗ has 18pq−p−q lines pq, where Degp=Degq=3.
From equation (21), we obtain(71)SDG∗=∑pq∈EG∗Degp2+Degq2Degp×DegqSDG∗=∑pq∈E1G∗Degp2+Degq2Degp×Degq+∑pq∈E2G∗Degp2+Degq2Degp×Degq+∑pq∈E3G∗Degp2+Degq2Degp×Degq=2E1G∗+135E2G∗+2E3G∗=212pq+4p+4q+13536pq−4q−4p+218pq−p−q.
From equation (22),(72)HG∗=∑pq∈EG∗2Degp×DegqHG∗=∑pq∈E1G∗2Degp×Degq+∑pq∈E2G∗2Degp×Degq+∑pq∈E3G∗2Degp×Degq=24E1G∗+25E2G∗+26E3G∗=1212pq+4p+4q+2536pq−4q−4p+1318pq−p−q=1325pq+1915p+1915q.


## 3. Conclusion

We find some topological descriptors of hydrocarbon structure. Randić index has well known applications in the study of physicochemical characteristics of alkane, for example, surface area, enthalpy of formation, boiling point, and melting point. The index has the most functional role in pharmacology. Similarly, forgotten index is helpful to figure out the strength of organic structures. Symmetric division index has productive applications to find heat formation of chemical structures. GA index can forecast biological activities of compounds. ABC index has an outstanding role in finding strain energy and stability of isoparaffins. Our results will be helpful to estimate the physicochemical properties of hydrocarbon structures.


## 4. Future Work

In future, we will find the distance-based and spectrum-based topological indices of hydrocarbon structures.



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*Source: 3621403-2022-01-05.xml*