# Computing Topological Indices and Polynomials of the Rhenium Trioxide

Authors: Shahid Imran; Muhammad Mudassar Raza; Niat Nigar; Syed Ajaz K. Kirmani; Fikre Bogale Petros
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the study of mathematical chemistry and chemical graph theory, a topological index, also known as a connectivity index, is the arithmetical framework of a graph that specifies its topology and also graph invariant. These topological indices are used to model quantitative structure relationshipsQSARs, which are connections between the work of biological or other molecular structures and the chemical structures. This study computed the first, second, and Hyper Zagreb indices, as well as Zagreb polynomials, Redefined Zagreb indices, Randic index, ABC index, and GA index of chemical structure of Rhenium Trioxide.


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## Body

## 1. Introduction

Graph theory is the branch of mathematics dealing with graphs which are basically the structures used to model pairwise relationship in different things. A graph consists of vertices that are affixed to edges [1]. A graph G is formed of VG≠ϕ of elements known as vertices, having finite numbers EG of unorganized pairings for VG units known as edges [2]. VG and EG were assigned to the vertex and edge sets of G, respectively. In chemical graph theory, atoms are represented by vertices and bonds are represented by edges [3, 4]. Chemical graph theory GT is the discipline of mathematical chemistry where we quantitatively implement devices of the graph premise to a material [5, 6]. Now a days, the proposal makes a significant contribution to computational sciences. As a result, we can define a subatomic diagram as a constrained network in which the nodes representing atoms and edges represent covalent bonds in concealed complex formation [7]. Symmetric key is a analytical standard associated to the composition that indicates the relationship between the concentration of the element and a wide range of physiosynthetic qualities, composite sensitivity, and organic action [8]. Molecular graphs are all chemical structures having structural formulas that comprise covalently bond compounds or molecules. Chemical graph theory tools were previously used for a variety of reasons, including numeration, systemization of the topic being discussed, and nomenclature [9, 10]. It also describes the process of assembling rules or regulations including a framework or strategy, as well as computer programming [11, 12]. The existence of isomerism, that is justified by the component graph hypothesis reinforces the necessity of graph theory techniques of science [13, 14]. The computational geometry of molecules according to well-defined rules is the essence of chemistry [15]. A topological graph index is a mathematical formula that may be applied to any graph that describes a molecule structure. The corresponding counting corresponding polynomials are some most common topological indices [16, 17]. On the other way, topological indices TIs are analytical principles related to chemical compositions for determining the relationship between chemical structure and various attributes. Physical qualities such as boiling temperatures, molar heats of formation, thermodynamic data, as well as chemical reactivity such as octane values as well as reactivity data, and biological activity are among these features [8, 18]. Topological indices, in particular, appear to hold promise for last mentioned goal, specifically QSAR [19, 20]. The quantitative structure-activity relationship QSAR is a computational modelling technique for determining connections between chemical compound structural features and biological activities [21]. In this research, we discuss about the inorganic compound ReO3 rhenium trioxide or rheniumVI oxide. It is a bright red solid with a metallic lustre that looks like copper. It is the Group 7th element, only stable trioxide. Having average estimation absorption of 1 part per billion, the most infrequent element on Earth crust is rhenium. From any stable elements, rhenium trioxide is the only element that has the 3rd highest melting point of 5903K. In this research, we talk about different topological indices and polynomials of rhenium trioxide.


## 2. Preliminaries

In the world of chemical graph theory, wiener indexWI which is known as wiener number presented by Henry wiener [22]. Actually, wiener index is the topological index of a molecule, stated nonhydrogen atoms in a molecule are denoted by total lengths of the shortest path in all pairs of vertices in a chemical graph.The number of edges linking a vertex determines its degree in graph theory. If′r′ and ′r′ are the vertices of a graph, then Ψt,Ψr are the degree of ′t′ and ′r′ vertices, respectively.Gutman and Trinajstic introduced in his study1st, 2nd Zegreb indices as(1)M1G=∑tr∈EGΨt+Ψr,M2G=∑tr∈EGΨt×Ψr.Shirdel et al. established “hyper Zagreb index” as follows:(2)HMG=∑tr∈EGΨt+Ψr2.Atom bond connectivity index is the most important and well known connectivity topological indices, presented by Estrada et al.:(3)ABCG=∑tr∈EGΨt+Ψr−2Ψt×Ψr.“Geometrically arithmetic indexGA” is another important topological figure of connectivity which is presented by Vukicevic et al.:(4)GAG=∑tr∈EG2Ψt×ΨrΨt+Ψr.Gutman and Trinajstic presented1st and 2nd Zagreb polynomial as(5)M1G,x=∑tr∈EGxΨt+Ψr,M2G,x=∑tr∈EGxΨt×Ψr.Redefined1st, 2nd, and 3rd Zagreb indices are the most significant indices presented by Usha et al. stated below as(6)ReZM1G=∑tr∈EGΨt+ΨrΨt×Ψr,ReZM2G=∑tr∈EGΨt×ΨrΨt+Ψr,ReZM3G=∑tr∈EGΨt+ΨrΨt×Ψr.There is also well known topological index called Randic in 1975 presented as(7)RIG=∑tr∈EG1Ψt×Ψr.SCI was introduced by Zhou et al. and defined as(8)SCIG=∑tr∈EG1Ψt+Ψr.


## 3. Main Results of Rhenium Trioxide

Rhenium is the derived from the Latin word Rhenus which means “Rhine” is the second last element that is discovered and having stable isotope. Rhenium is a chemical element that is denoted by′Re′ with atomic number 75. It is silver gray colored, heavy, and 3rd row transition metal in 7th group of periodic table. Rhenium look like manganese and technetium chemically in nature; it is acquired by the consequence of ancestry and purification copper ores and molybdenum. Rhenium expresses the number of oxidation states in the wide limitation of −1 to +7.As compared others, the oxide ReO3 is the best electrical conductor at room temperature resistively that is close to copper, as shown in Figure 1. The cubic crystal of ReO3 identifies particular perovskite-type crystal structure that is related to O1hPm3m having lattice parameter a0=3.7504A0. The crystal lattice of ReO6 contains octahedra joined from edges and the unit cell of Bravais that holds f.u. of ReO3. The red colored rhenium trioxide shows metallic conductivity, it is also called covalent metal (see Figure 1).Figure 1
Structure of rhenium trioxide.Theorem 1.
IfR is the graph of rhenium trioxide as illustrated in Figure 1, then(9)M1R=154mn+44m−26n+20,M2R=366mn+87m−63n,HMR=1502mn+352m−410n,PM1R=218mn−3m−9n+10×34mn+8m+8n−16×56mn+3m−3n−2×72m+2n×114mn−2m−2n,PM2R=26mn+16m+16n×312mn×524mn+2m−10n−12,M1G,x=2m+2nx7+4mn−2m−2n+4x8+2mn+4m+4n−8x9+6mn+3m−3n−2x10+4mn−2m−2nx11,M2G,x=2m+2nx12+4mn−2m−2nx15+4x16+2mnx18+4m+4m−8x20+2m+2nx24+6mn+m−5n−2x25+4mn−2m−2nx30,ReZM1R=275mn+2110m−310n−125,ReZM2R=82322mn+10.4129m−4.5871n−1339,ReZM3R=3624mn+718m−782n−1428,RIR=3.4345mn+1.1985m−0.0015n−1.6833,SCIR=5.1843mn+1.7278m−0.1696n−1.8849,HIR=1669495mn+69076930m−236930n−5345,ABCR=9.3620mn+3.0175m−0.3767n−3.4147,GAR=15.7420mn+4.9861m−1.0139n−5.9505,Proof.
LetG be isomorphic to rhenium trioxide. Then, the edge partition for the chemical graph of rhenium trioxide based on degree sum of vertices of each other is given by(10)E34=2m+n=2m+2n,E35=4mn−2m−2n,E36=2mn,E44=4,E45=4m+4n−8,E46=2m+n=2m+2n,E55=6m+m−5n−2m<2,n<5,mn∈Z+,E56=4mn−2m−2n.


### 3.1. First Zagreb Index

(11)M1G=∑tr∈EGΨt+Ψr,M1R=∑tr∈E34RΨt+Ψr+∑tr∈E35RΨt+Ψr+∑tr∈E36RΨt+Ψr+∑tr∈E44RΨt+Ψr+∑tr∈E45RΨt+Ψr+∑tr∈E46RΨt+Ψr+∑tr∈E55RΨt+Ψr+∑tr∈E56RΨt+Ψr=7E34R+8E35R+9E36R+8E44R+9E45R+10E46R+10E55R+11E56R=72m+n+84mn−2m−2n+92mn+84+94m+4n−8+102m+n+106mn+m−5n−2+114mn−2m−2n,M1R=154mn+44m−26n+20.


### 3.2. Second Zagreb Index

(12)M2G=∑tr∈EGΨt×Ψr=12E34R+15E35R+18E36R+16E44R+20E45R+24E46R+25E55R+30E56R=122m+n+154mn−2m−2n+182mn+164+204m+4n−8+242m+n+256mn+m−5n−2+304mn−2m−2n,M2R=366mn+87m−63n.


### 3.3. Hyper Zagreb Index

(13)HM1G=∑tr∈EGΨt+Ψr2=49E34R+64E35R+81E36R+64E44R+81E45R+100E46R+100E55R+121E56R=492m+n+644mn−2m−2n+812mn+644+814m+4n−8+1002m+n+1006mn+m−5n−2+1214mn−2m−2n,HM1R=1502mn+352m−410n.The comparison betweenM1G, M2G, and HMG is shown in Table 1 and Figure 2.Table 1
Comparison betweenM1G, M2G, and HMG.
m,nM1RM2RHMR1,119239014442,2672151258923,314603366133444,425565952238005,539609270372606,6567213320537247,7769218102731928,81002023616956649,9126562986212114010,101560036840149620Figure 2
The graphical representation of Table1.


### 3.4. First Zagreb Polynomial

(14)M1G1,x=∑tr∈EGxΨt+Ψr=E34Rx7+E35Rx8+E36Rx9+E44Rx8+E45Rx9+E46Rx10+E55Rx10+E56Rx11,M1R1,x=2m+2nx7+4mn−2m−2nx8+2mnx9+4x8+4m+4n−8x9+2m+2nx10+6mn+m−5n−2x10+4mn−2m−2nx11+6mn+3m−3n−2x10+4mn−2m−2nx11.


### 3.5. Second Zagreb Polynomial

(15)M2G1,x=∑tr∈EGxΨt×Ψr=E34Rx12+E35Rx15+E36Rx18+E44Rx16+E45Rx20+E46Rx24+E55Rx25+E56Rx30,M2R1,x=2m+2nx12+4mn−2m−2nx15+2mnx18+4x16+4m+4n−8x20+2m+2nx24+6mn+m−5n−2x25+4mn−2m−2nx30.The comparison between first and second Zagreb polynomials is represented in Table2 and Figure 3.Table 2
Connection betweenM1G1,x and M2G1,x.
m,nM1G1,xM2G1,x1,1,114142,2,166663,3,11501504,4,12662665,5,14144146,6,15945947,7,18068068,8,1105010509,9,11326132610,10,116341634Figure 3
The graphical representation of Table2.


### 3.6. Redefined First Zagreb Index

(16)ReZM1G=∑tr∈EGΨt+ΨrΨt×Ψr=712E34R+815E35R+918E36R+816E44R+920E45R+1024E46R+1025E55R+1130E56R=7122m+n+8154mn−2m−2n+122mn+124+9204m+4n−8+5122m+n+256mn+m−5n−2+11304mn−2m−2n,ReZM1R=275mn+2110m−310n−125.


### 3.7. Redefined Second Zagreb Index

(17)ReZM2G=∑tr∈EGΨt×ΨrΨt+Ψr=127E34R+158E35R+189E36R+168E44R+209E45R+2410E46R+2510E55R+3011E56R=1272m+n+1584mn−2m−2n+1892mn+1684+2094m+4n−8+24102m+n+25106mn+m−5n−2+30114mn−2m−2n,ReZM2R=82322mn+10.4129m−4.5871n−1339.


### 3.8. Redefined Third Zagreb Index

(18)ReZM3G=∑tr∈EGΨt+ΨrΨt×Ψr=7×12E34R+8×15E35R+9×18E36R+8×16E44R+9×20E45R+10×24E46R+10×25E55R+11×30E56R=842m+n+1204mn−2m−2n+1622mn+1284+1804m+4n−8+2402m+n+2506mn+m−5n−2+3304mn−2m−2n,ReZM3R=3624mn+718m−782n−1428.The comparison between redefined Zagreb indices is represented in Table3 and Figure 4.Table 3
Connection betweenReZG1, ReZG2, and ReZG3.
m,nReZG1GReZG2GReZG3G1,14.828.457121322,228.2146.5102129403,351.6339.3814309964,491.2607.0709563005,5141.6949.5785888526,6202.81366.90431286527,7274.81859.04831757008,8357.62426.01042299969,9451.23067.790829154010,10555.63784.3893360332Figure 4
Graphical representation of Table3.


### 3.9. Randic Index

(19)RIG=∑tr∈EG1Ψt×Ψr=1122m+n+1154mn−2m−2n+1182mn+1164+1204m+4n−8+1242m+n+1256mn+m−5n−2+1304mn−2m−2n,RIR=3.4345mn+1.1985m−0.0015n−1.6833.


### 3.10. Sum Connectivity Index

(20)SCIG=∑tr∈EG1Ψt+Ψr=172m+n+184mn−2m−2n+132mn+1224+134m+4n−8+1102m+n+1106mn+m−5n−2+1114mn−2m−2n,SCIR=5.1843mn+1.7278m−0.1696n−1.8849.


### 3.11. Harmonic Index

(21)HIG=∑tr∈EG2Ψt×Ψr272m+n+144mn−2m−2n+292mn+144+294m+4n−8+2102m+n+156mn+m−5n−2+2114mn−2m−2n,HIR=1669495mn+69076930m−236930n−5345.The connection between theRIR, SCIR, and HIR is shown in Table 4 and Figure 5.Table 4
Connection betweenRIR, SCIR, and HIR.
m,nRIRSCIRHIR1,12.94824.85763.18732,214.448721.968714.29583,332.818249.448432.14784,448.468787.296756.74315,590.1642135.513688.08206,6129.1407194.0991126.16427,7167.9862263.0532170.98998,8228.2157342.3759222.55909,9287.2842432.0672280.871610,10353.7367532.1271345.9276Figure 5
Graphical representation of Table4.


### 3.12.ABC Index

(22)ABCG=∑tr∈EGΨt+Ψr−2Ψt×Ψr=15162m+n+1054mn−2m−2n+1462mn+644+35104m+4n−8+332m+n+2256mn+m−5n−2+30104mn−2m−2n,ABCR=9.3620mn+3.0175m−0.3767n−3.4147.


### 3.13. GA Index

(23)GAG=∑tr∈EG2Ψt×ΨrΨt+Ψr=4372m+n+1544mn−2m−2n+2232mn+14+4594m+4n−8+2652m+n+16mn+m−5n−2+230114mn−2m−2n,GAR=15.7420mn+4.9861m−1.0139n−5.9505.The connection between ABCR and GAR is shown in Table 5 and Figure 6.Table 5
Connection betweenABCR and GAR
m,nABCRGAR1,18.588113.76372,239.314964.96193,388.7657147.64414,4156.9405261.81035,5243.8393407.46056,6349.4621584.59477,7473.8089793.21298,8616.87971033.31519,9778.67451304.901310,10959.19331607.9715Figure 6
The graphical representation of Table5.


## 3.1. First Zagreb Index

(11)M1G=∑tr∈EGΨt+Ψr,M1R=∑tr∈E34RΨt+Ψr+∑tr∈E35RΨt+Ψr+∑tr∈E36RΨt+Ψr+∑tr∈E44RΨt+Ψr+∑tr∈E45RΨt+Ψr+∑tr∈E46RΨt+Ψr+∑tr∈E55RΨt+Ψr+∑tr∈E56RΨt+Ψr=7E34R+8E35R+9E36R+8E44R+9E45R+10E46R+10E55R+11E56R=72m+n+84mn−2m−2n+92mn+84+94m+4n−8+102m+n+106mn+m−5n−2+114mn−2m−2n,M1R=154mn+44m−26n+20.


## 3.2. Second Zagreb Index

(12)M2G=∑tr∈EGΨt×Ψr=12E34R+15E35R+18E36R+16E44R+20E45R+24E46R+25E55R+30E56R=122m+n+154mn−2m−2n+182mn+164+204m+4n−8+242m+n+256mn+m−5n−2+304mn−2m−2n,M2R=366mn+87m−63n.


## 3.3. Hyper Zagreb Index

(13)HM1G=∑tr∈EGΨt+Ψr2=49E34R+64E35R+81E36R+64E44R+81E45R+100E46R+100E55R+121E56R=492m+n+644mn−2m−2n+812mn+644+814m+4n−8+1002m+n+1006mn+m−5n−2+1214mn−2m−2n,HM1R=1502mn+352m−410n.The comparison betweenM1G, M2G, and HMG is shown in Table 1 and Figure 2.Table 1
Comparison betweenM1G, M2G, and HMG.
m,nM1RM2RHMR1,119239014442,2672151258923,314603366133444,425565952238005,539609270372606,6567213320537247,7769218102731928,81002023616956649,9126562986212114010,101560036840149620Figure 2
The graphical representation of Table1.


## 3.4. First Zagreb Polynomial

(14)M1G1,x=∑tr∈EGxΨt+Ψr=E34Rx7+E35Rx8+E36Rx9+E44Rx8+E45Rx9+E46Rx10+E55Rx10+E56Rx11,M1R1,x=2m+2nx7+4mn−2m−2nx8+2mnx9+4x8+4m+4n−8x9+2m+2nx10+6mn+m−5n−2x10+4mn−2m−2nx11+6mn+3m−3n−2x10+4mn−2m−2nx11.


## 3.5. Second Zagreb Polynomial

(15)M2G1,x=∑tr∈EGxΨt×Ψr=E34Rx12+E35Rx15+E36Rx18+E44Rx16+E45Rx20+E46Rx24+E55Rx25+E56Rx30,M2R1,x=2m+2nx12+4mn−2m−2nx15+2mnx18+4x16+4m+4n−8x20+2m+2nx24+6mn+m−5n−2x25+4mn−2m−2nx30.The comparison between first and second Zagreb polynomials is represented in Table2 and Figure 3.Table 2
Connection betweenM1G1,x and M2G1,x.
m,nM1G1,xM2G1,x1,1,114142,2,166663,3,11501504,4,12662665,5,14144146,6,15945947,7,18068068,8,1105010509,9,11326132610,10,116341634Figure 3
The graphical representation of Table2.


## 3.6. Redefined First Zagreb Index

(16)ReZM1G=∑tr∈EGΨt+ΨrΨt×Ψr=712E34R+815E35R+918E36R+816E44R+920E45R+1024E46R+1025E55R+1130E56R=7122m+n+8154mn−2m−2n+122mn+124+9204m+4n−8+5122m+n+256mn+m−5n−2+11304mn−2m−2n,ReZM1R=275mn+2110m−310n−125.


## 3.7. Redefined Second Zagreb Index

(17)ReZM2G=∑tr∈EGΨt×ΨrΨt+Ψr=127E34R+158E35R+189E36R+168E44R+209E45R+2410E46R+2510E55R+3011E56R=1272m+n+1584mn−2m−2n+1892mn+1684+2094m+4n−8+24102m+n+25106mn+m−5n−2+30114mn−2m−2n,ReZM2R=82322mn+10.4129m−4.5871n−1339.


## 3.8. Redefined Third Zagreb Index

(18)ReZM3G=∑tr∈EGΨt+ΨrΨt×Ψr=7×12E34R+8×15E35R+9×18E36R+8×16E44R+9×20E45R+10×24E46R+10×25E55R+11×30E56R=842m+n+1204mn−2m−2n+1622mn+1284+1804m+4n−8+2402m+n+2506mn+m−5n−2+3304mn−2m−2n,ReZM3R=3624mn+718m−782n−1428.The comparison between redefined Zagreb indices is represented in Table3 and Figure 4.Table 3
Connection betweenReZG1, ReZG2, and ReZG3.
m,nReZG1GReZG2GReZG3G1,14.828.457121322,228.2146.5102129403,351.6339.3814309964,491.2607.0709563005,5141.6949.5785888526,6202.81366.90431286527,7274.81859.04831757008,8357.62426.01042299969,9451.23067.790829154010,10555.63784.3893360332Figure 4
Graphical representation of Table3.


## 3.9. Randic Index

(19)RIG=∑tr∈EG1Ψt×Ψr=1122m+n+1154mn−2m−2n+1182mn+1164+1204m+4n−8+1242m+n+1256mn+m−5n−2+1304mn−2m−2n,RIR=3.4345mn+1.1985m−0.0015n−1.6833.


## 3.10. Sum Connectivity Index

(20)SCIG=∑tr∈EG1Ψt+Ψr=172m+n+184mn−2m−2n+132mn+1224+134m+4n−8+1102m+n+1106mn+m−5n−2+1114mn−2m−2n,SCIR=5.1843mn+1.7278m−0.1696n−1.8849.


## 3.11. Harmonic Index

(21)HIG=∑tr∈EG2Ψt×Ψr272m+n+144mn−2m−2n+292mn+144+294m+4n−8+2102m+n+156mn+m−5n−2+2114mn−2m−2n,HIR=1669495mn+69076930m−236930n−5345.The connection between theRIR, SCIR, and HIR is shown in Table 4 and Figure 5.Table 4
Connection betweenRIR, SCIR, and HIR.
m,nRIRSCIRHIR1,12.94824.85763.18732,214.448721.968714.29583,332.818249.448432.14784,448.468787.296756.74315,590.1642135.513688.08206,6129.1407194.0991126.16427,7167.9862263.0532170.98998,8228.2157342.3759222.55909,9287.2842432.0672280.871610,10353.7367532.1271345.9276Figure 5
Graphical representation of Table4.


## 3.12.ABC Index

(22)ABCG=∑tr∈EGΨt+Ψr−2Ψt×Ψr=15162m+n+1054mn−2m−2n+1462mn+644+35104m+4n−8+332m+n+2256mn+m−5n−2+30104mn−2m−2n,ABCR=9.3620mn+3.0175m−0.3767n−3.4147.


## 3.13. GA Index

(23)GAG=∑tr∈EG2Ψt×ΨrΨt+Ψr=4372m+n+1544mn−2m−2n+2232mn+14+4594m+4n−8+2652m+n+16mn+m−5n−2+230114mn−2m−2n,GAR=15.7420mn+4.9861m−1.0139n−5.9505.The connection between ABCR and GAR is shown in Table 5 and Figure 6.Table 5
Connection betweenABCR and GAR
m,nABCRGAR1,18.588113.76372,239.314964.96193,388.7657147.64414,4156.9405261.81035,5243.8393407.46056,6349.4621584.59477,7473.8089793.21298,8616.87971033.31519,9778.67451304.901310,10959.19331607.9715Figure 6
The graphical representation of Table5.


## 4. Conclusion

In this research work, researchers computed different topological indices such as Zagreb indices, redefined Zagreb indices, atom bond connectivity, harmonic index and sum connectivity index, and many other indices of chemical graph of rhenium trioxideReO3. Rhenium trioxide also has been expanded by a researcher upto m and n cycles. Researchers also explained the comparison between the different versions of topological indices in a numerical way with the help of table and also expressed them in the graphical pattern. By comparing these topological indices with tables and graphs, the importance of indices shows that these indices results have significant relationship with each other.



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