# Topological Study of Zirconium TetrachlorideZrCl4 under Molecular Descriptors

Authors: Zeeshan Saleem Mufti; Rukhshanda Anjum; Fairouz Tchier; Hafsa Sajid; Qin Xin; Faria Ahmed Shami
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Topological index is a numerical parameter which characterizes the topology of the molecular structure. Topological indices are a very prominent part of the study of chemical structures in which properties of organic or inorganic compounds are under observation and calculated such as physical properties, chemical reactivity, or biological activity. Most of the topological indices of molecular graph-based structure which depends on vertex degrees have been visualized. In this study, we compute some degree-based topological indices of zirconium tetrachlorideZnCl4m,n.


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## Body

## 1. Introduction

Graph theory is intimately linked to the information technology, engineering, optimization theory, web designing, economics, biology, and chemistry. Topological index of a chemical compound is an integer which can be used to characterize the topology of the atomic frame and predicts some physicochemical properties. Nowadays, an organic investigation of chemical mixtures is high priced and demands modern apparatus to investigate these mixtures in laboratory. This exercise is expensive and slow [1, 2]. As a result of this aspect, to observe the advance inventions and methods by which the market price could be minimized, the patent medicine companies are very committed. By studying the obvious atomic frames using topological indices, one can minimize the market price without any use of laboratories and equipment [1, 3–9].In this analysis, we use some topological indices of molecular graph zirconium tetrachlorideZrCl4 and correlate with the physicochemical properties such as heat of formation, enthalpy, and infrared spectrum. That influenced us to work on the chemical structure of zirconium tetrachloride ZrCl4m,n under this phenomenon [10–15].Topological indices are used in QSAR and QSPR analysis as an application. Numerous applications of different topological indices such as Randic index is being used in cheminformatics to analyse the organic compounds and ABC index is used to study physicochemical properties of the hydrocarbons structures.A molecular graphG=V˜,E˜ is a simple graph with the order n=V˜ and the size m=E˜. The order of a graph, denoted by n, is the number of vertices in the graph and the size, denoted by n, is the number of edges in a graph. Let dv represent the degree of the vertex v, and it is defined as the total count of edges connected to v. Let Sv represent the degree sum of all its neighbors [16]. For basics of graphs and topological indices, see [2, 17–23].Here, we discuss some degree-based topological indices to compute the values of zirconium tetrachlorideZrCl4m,n network.The general Randic connectivity index ofG˜ is(1)RαG=∑rs∈E˜G˜drdsα,where α represents a real number. The ABC index emerged by Estrada and Torres. The ABC index is(2)ABCG=∑rs∈E˜Gdr+ds−2drds.Vukicevic and Furtula announced the geometric-arithmetic (GA) index which is denoted by GA:(3)GAG=∑rs∈E˜G2drdsdr+ds.The forgotten index F(G) was investigated by Furtula and Gutman which is defined as(4)FG=∑rs∈E˜Gdr2+ds2.The second Zagreb index has been invented by Gutman and Trinajestic:(5)M2G=∑rs∈E˜Gdr×ds.Furtula et al. also introduced augmented Zagreb index:(6)AZIG=∑rs∈E˜Gdrdsdr+ds−23.The invariant reciprocal Randic index RR has been introduced first time by Favaron and Mah’eo. It is explained as(7)RRG=∑rs∈E˜Gdrds.The reduced reciprocal Randic index may be observed as the decreased model of the reciprocal Randic index, 7:(8)RRRG=∑rs∈E˜Gdr−1×ds−1.Another index named forth class of (ABC) has been presented by Ghorbani et al. in 2010 [24, 25] as(9)ABC4G=∑rs∈E˜GSr+Ss−2SrSs.The fifth class of geometric-arithmetic index, indicated byGA5, was introduced by Graovac et al. in 2011 [26] as(10)GA5G=∑rs∈E˜G2SrSsSr+Ss.


## 2. Computing the Topological Indices of Zirconium TetrachlorideZrCl4m,n Graph

In this study, we will use a new graph called zirconium tetrachloride. We get the zirconium tetrachloride from the reaction of zirconium metal and chlorine gas at300°C. The high dissolvability in the residence of methylated benzene is the most strange property of ZrCl4m,n, where m represents the horizontal unit and n represents the vertical unit. It is an inorganic compound regularly used as an antecedent to other compounds of zirconium. This white high melting solid hydrolyzes quickly in the human air.It is also used to make water-repellent treatment of textiles and other fibrous materials. Zirconium tetrachloride is also known as zirconium chloride which is shown in Figure1. There are 2mn+8m+8n+30 vertices and 6mn+20m+20n+65 edges.Figure 1
Graph of zirconium tetrachloride (5, 4).Theorem 1.
LetG be the graph of zirconium tetrachloride ZrCl4m,n; then, its Randic index is(11)RαG=n+78α+2m+n+714α+2m+n+712α+m+2n+521α+3mn+6m+6n+1227α+16α+n+528α+2m+n+736α+m+249α+2m+3n+763α+3mn+4m+4n+581α.Proof.
The network of zirconium tetrachloride has6mn+20m+20n+65 number of edges, where m,n>3. The edge set E˜G of G is split up in eleven subdivisions based upon degrees of end vertices, i.e., E˜G=E˜1G∪E˜2G∪E˜3G∪E˜4G∪E˜5G∪E˜6G∪E˜7G∪E˜8G∪E˜9G∪E˜10G∪E˜11G. By using Table 1 and formula (1), we gain(12)RαG=8αE˜1G+14αE˜2G+12αE˜3G+21αE˜4G+27αE˜5G+16αE˜6G+28αE˜7G+36αE˜8G+49αE˜9G+63αE˜10GG+81αE˜11G.
After some calculations, we receive the specific conclusion.Table 1
Edge separation of zirconium tetrachloride on account of starting and ending vertices of each edge.
dr,dsEdgesdr,dsEdges(2, 4)n + 7(2, 7)2m + n+7(3, 4)2m + n+7(3, 7)m+2n+5(3, 9)3mn + 6m + 6n + 12(4, 4)1(4, 7)n + 5(4, 9)2m + n+7(7, 7)m + 2(7, 9)2m + 3n + 7(9, 9)3mn + 4m + 4n + 5——Total edges6mn + 20m + 20n + 65Theorem 2.
LetG be the graph of zirconium tetrachloride ZrCl4m,n; its atom-bond connectivity index is(13)ABCG=12n+72+122m+n+72+162m+n+715+221m+2n+542+193mn+6m+6n+1230+146+314n+57+162m+n+711+27m+23+132m+3n+72+43mn+169m+169n+209.Proof.
The network of zirconium tetrachloride has6mn+20m+20n+65 number of edges, where m,n>3. The edge set E˜G of G is split up in eleven subdivisions based on degrees of end vertices. By using Table 1 and formula (2), we gain(14)ABCG=2+4−22×4E˜1G+2+7−22×7E˜2G+3+4−23×4E˜3G+3+7−23×7E˜4G+3+9−23×9E˜5G+4+4−24×4E˜6G+4+7−24×7E˜7G+4+9−24×9E˜8G+7+7−27×7E˜9G+7+9−27×9E˜10G+9+9−29×9E˜11G.
After some calculations, we receive the specific conclusion.Theorem 3.
LetG be the graph of zirconium tetrachloride ZrCl4m,n; its geometric-arithmetic index is(15)GAG=23n+72+292m+n+714+472m+n+73+15m+2n+521+123mn+6m+6n+123+18813+411n+57+8913m+6413n+382m+3n+77+3mn.Proof.
The network of zirconium tetrachloride has6mn+20m+20n+65 number of edges, where m,n>3. The edge set E˜G of G is split up in eleven subdivisions based on degrees of end vertices. By using Table 1 and formula (3), we gain(16)GAG=22×42+4E˜1G+22×72+7E˜2G+23×43+4E˜3G+23×73+7E˜4G+23×93+9E˜5G+24×44+4E˜6G+24×74+7E˜7G+24×94+9E˜8G+27×77+7E˜9G+27×97+9E˜10G+29×99+9E˜11G.
After some calculations, we receive the specific conclusion.Theorem 4.
LetG be the graph of zirconium tetrachloride ZrCl4m,n. Then,(17)RM2G=2721+n3+808m+814n+240mn.Proof.
The network of zirconium tetrachloride has6mn+20m+20n+65 number of edges, where m,n>3. The term er,s is the edge between the vertices r and s. There are 11 partitions of the edge sets of the graphs. The first partition set is e2,4, the second partition set is e2,7, the third partition set is e3,4, the fourth partition set is e3,7, the fifth partition set is e3,9, the sixth partition set is e4,4, the seventh partition set is e4,7, the eighth partition set is e4,9, the ninth partition set is e7,7, the tenth partition set is e7,9 and the eleventh partition set is e9,9. By using Table 1 and formula (5), we gain(18)RM2G=e2,42−14−1+e2,72−17−1+e3,43−14−1+e3,73−17−1+e3,93−19−1+e4,44−14−1+e4,74−17−1+e4,94−19−1+e7,77−17−1+e7,97−19−1+e9,99−19−1.
After some calculations, we receive the specific conclusion.Theorem 5.
LetG be the graph of zirconium tetrachloride ZrCl4m,n. Then,(19)AZIG=229523463512000mn+54005009664713679948800m+740177969760073496793088000n+5053056956206110349856000.Proof.
The network of zirconium tetrachloride has6mn+20m+20n+65 number of edges, where m,n>3. By using Table 1 and formula (6), we gain(20)AZIG=e2,42×42+4−23+e2,72×72+7−23+e3,43×43+4−23+e3,73×73+7−23+e3,93×93+9−23+e4,44×44+4−23+e4,74×74+7−23+e4,94×94+9−23+e7,77×77+7−23+e7,97×97+9−23+e9,99×99+9−23.
After some calculations, we receive the specific conclusion.Theorem 6.
LetG be the graph of zirconium tetrachloride ZrCl4m,n. Then,(21)1RRG=2n+72+2m+n+714+22m+n+73+m+2n+521+33mn+6m+6n+123+105+2n+57+55m+42n+32m+3n+77+27mn,2RRRG=n+73+42m+n+76+2m+2n+53+36mn+62m+56n+103+3n+52+42m+3n+73.Proof.
The network of zirconium tetrachloride has6mn+20m+20n+65 number of edges, where m,n>3.(1)
By using Table1 and formula (7), we gain(22)RRG=e2,42×4+e2,72×7+e3,43×4+e3,73×7+e3,93×9+e4,44×4+e4,74×7+e4,94×9+e7,77×7+e7,97×9+e9,99×9.After some calculations, we receive the specific conclusion.(2)
By using Table1 and formula (8), we gain(23)RRRG=e2,42−1×4−1+e2,72−1×7−1+e3,43−1×4−1+e3,73−1×7−1+e3,93−1×9−1+e4,44−1×4−1+e4,74−1×7−1+e4,94−1×9−1+e7,77−1×7−1+e7,97−1×9−1+e9,99−1×9−1.
After some calculations, we receive the specific conclusion.Theorem 7.
LetG be the graph of zirconium tetrachloride ZrCl4m,n. Then,(24)FG=756mn+1954m+1954n+5008.Proof.
The network of zirconium tetrachloride has6mn+20m+20n+65 number of edges, where m,n>3. By using Table 1 and formula (4), we gain(25)FG=e2,422+42+e2,722+72+e3,432+42+e3,732+72+e3,932+92+e4,442+42+e4,742+72+e4,942+92+e7,772+72+e7,972+92+e9,992+92.
After some calculations, we receive the specific conclusion.Now, we findABC4G and GA5G indices (Table 2) of zirconium tetrachloride ZnCl4m,n.Table 2
Edge separation of zirconium tetrachloride on account of degree sum of neighboring ending vertices of each edge.
dr,dsNeighboring verticesdr,dsNeighboring vertices(8, 18)2(11, 18)2(11, 21)n + 3(11, 33)2(11, 34)1(11, 36)1(11, 37)1(11, 39)n(14, 34)1(14, 36)1(14, 39)2m(17, 18)1(17, 21)3(17, 24)2m + 2(17, 46)1(17, 49)1(17, 51)1(17, 53)m(18, 18)1(18, 20)1(18, 34)1(18, 37)1(18, 46)2(20, 21)n(20, 37)1(20, 39)n(20, 46)1(20, 54)n(21, 33)2(21, 36)1(21, 39)n(21, 49)2(21, 51)1(21, 54)n(23, 34)1(23, 37)1(23, 57)1(24, 46)1(24, 51)1(24, 53)2m(25, 33)1(25, 36)1(25, 39)m + n(25, 49)2(25, 51)1(25, 54)n(25, 57)1(25, 59)2m(25, 61)n(27, 46)1(27, 51)1(27, 53)2m(27, 54)n(27, 57)1(27, 59)m(27, 61)2n(27, 63)3mn(33, 51)1(33, 54)1(34, 37)1(34, 39)1(34, 57)1(36, 39)1(36, 49)1(36, 59)1(37, 54)1(37, 57)1(39, 39)m − 1(39, 46)1(39, 49)1(39, 54)2n − 2(39, 57)1(39, 59)2m − 1(39, 61)n(46, 53)1(46, 61)1(49, 59)1(49, 61)1(51, 53)1(51, 54)1(51, 63)1(53, 53)m − 1(53, 61)1(53, 63)2m − 1(54, 57)1(54, 61)2n − 2(54, 63)n(57, 59)1(57, 63)1(59, 59)m − 1(59, 61)1(59, 63)2m − 1(61, 63)3n − 2(63, 63)3mn − 2m − 2n + 1Total vertices6mn + 20m + 20n + 65Theorem 8.
LetG be the graph of zirconium tetrachloride ZrCl4m,n; then, its fourth atom-bond connectivity index is(26)ABC4G=−7+159118+161122+136+11166+137416082+140718722+12385474+12142+11021122+178247702+11010+11794148902+11938172482+1132694146+1125886802+143711362+185149358+178243010+12806294630+12438236486+1378n4830+2633mn−2m−2n+131+1213n−214+212392m−112390+1682m+2442+253m53+1130n1235+115n15+1182n−214+1273n5278+177n+3770+1126n1022+1195m+n2418+190n462+2305n1281+263mn462+191m3094+2549n15738+2159m1378+5106m106+2295m4838+21114+2165462+153106+72379n4758+253m26+259m29+24767m1534+415351+1119747082+12255+151170+4143n143+1942499357+811917+11722+1615334+12223922+1691426+174123218+239m19+2210948507+166619758+13545487+13063502+14283+12342847+211098n41358+11981870+156115334+2177m413+11711558+154n158+6427183+2153323+14149798+12733354+4285285+14136254+12553774+12352+13059+12042482+11381173+170n455+174407+13424142+2115230+42311001+211113m42294+11261155+4323322631.Proof.
The network of zirconium tetrachloride has6mn+20m+20n+65 number of edges, where m,n>3. There are 94 disconnected degree sum of neighboring ending vertices of each edge, i.e., e8,18,e11,18,e11,21,e11,33,e11,34,e11,36,e11,3,e11,39,⋅⋅⋅⋅⋅,e63,63. By using Table 2 and formula (4), we receive specific conclusion.Theorem 9.
Consider the zirconium tetrachlorideZrCl4 network; its fifth geometric-arithmetic index is(27)GA5G=61910+323+62514+53851+419357+28942917+182922+8817534+m+1313513059+124407+233230+1401311+130851+801197782+245374+877m318+532m+n39+625n14+110n91+459n195+441n105+637n30+4412m+2102+3612m−1413+35mn21+322n183+521m59+320m159+61152n−2366+23n2+543n61+3079n6+3582m−1371+213n42+6312n−226+3623n−2427+135m901+116n+3231+453m546+1603599+1583363+150n2379+1492m−12301+1573233+1522703+21072806+2992438+1472109+2851794+2911938+2731326+2711258+960219m177+117463892493985+124711+43569+16119+2977+457185+3823+65574+114187+673138+54157+52933+691222+116247+75561+63738+120399+589521+n125429+351110039−2n+3mn.Proof.
The network of zirconium tetrachloride has6mn+20m+20n+65 number of edges, where m,n>3. There are 94 disconnected degree sum of neighboring ending vertices of each edge, i.e., e8,18,e11,18,e11,21,e11,33,e11,34,e11,36,e11,3,e11,39,⋅⋅⋅⋅⋅,e63,63. By using Table 2 and formula (4), we receive specific conclusion.


## 3. Conclusion

We have determined some topological indices of zirconium tetrachlorideZrCl4 structure. We have explored the network hypothetically not in laboratory. Many problems have been solved using topological indices of different structures from different branches of sciences such as computer science, chemistry, and information technology. Randic index is used in cheminformatics to study chemical compounds. Similarly, ABC and ABC4 are used to check the stability of the chemical compound. In this study, the acquired results come up with the remarkable benefaction in graph theory and tally the chemical structure of zirconium tetrachloride ZrCl4m,n with lots of facts about physicochemical properties such as freezing points, boiling points, melting points, infrared spectrum, electronic parameters, viscosity, molecular weight, and density. In future, we will find the topological indices of more complex structures such as line and para-line graph of the zirconium tetrachloride ZrCl4m,n structure [27, 28].



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