# On Neighborhood Degree-Based Topological Analysis of Polyphenylene Network

Authors: Chuang Sun; A. Khalid; H. M. Usman; A. Ahmad; M. K. Siddiqui; S. A. Fufa
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Organic compounds such as polyphenylene are very important and useful for the synthesis of many new organic compounds due to their physio-chemical properties. To ascertain these properties, one can use QSPR/QSAR methods which necessitate the computation of topological indices. The topological indices based on two newly introduced abstract notions of ev-degree and ve-degree are in practice to model numerous chemical properties as well as physical properties of organic, inorganic, hybrid, and biological compounds. In this study, we computed a certain number of topological indices for the chemical graph of polyphenylene network which will help to model some of its physio-chemical properties.


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## Body

## 1. Introduction

The detailed critical inspection in order to discover essential features or meanings of chemical compounds graphically is known as chemical graph theory. It is the branch of mathematics, which alloys chemistry and graph theory. In graph theory, a simple graph or just graphGV,E is constructed by two sets: V=v1,…,vn, the set of vertices, and E=e1,…,em, the set of edges. Each v∈V represents a node in the graph and each e∈E denotes the line joining two nodes.In chemical graph theory, the image obtained from diffraction of X-rays or electron microscopy of a compound (biological or chemical) is drawn into plane and lighted upon its symmetry, and then, peculiarities of this compound is mathematically modeled. The simple sketch of the image of compound is known as the chemical graph where we assume that the ends or vertices are atoms and lines or edges are the bonds between the atoms. Chemical graph theory helps to understand different properties, namely, molecular structure, kinetics of molecules, atoms or electrons, chain or patterns of polymers, crystals and clusters, aromaticity, nuclear magnetic resonance (NMR) analysis, depicting orbitals, and electrons behaviors. Ante Graovac, Alexandru Balaban, Haruo Hosoya, Iva′n Gutman, Nenad Trinajstic, and Milan Randic are few scientists who introduced graph theory in chemistry [1].The job of mathematical modeling the properties of chemical compounds is done by topological indices which we define asa number obtained by a real-valued function,g=ge,g=gvorg=ge,v, that is applied to any chemical graph (or molecular structure) of a compound to determine its topology, is known as topological graph index or just topological index (plural: topological indices), whereeandv∈ℤ+are edges and vertices of graph, for example, Zagreb indices and their variants, distance indices, detour index, and Wiener index. There are different kinds of topological indices based on degree, distance, and counting [2]. Many physical and chemical properties of different chemical and biological compounds have been modeled mathematically by the aid of topological indices such as boiling point, anti-leishmanial effect, acute toxicity, radial scavenging activity, and many more [3–5]. In this study, we considered some topological indices based on degree of vertices and edges, ve-degree of vertices, and ev-degree of edges. To understand the terms used in formulas of topological indices, first, we consider the following basic definitions [6–8].For a graphXV,E, we have(1)
Degree of a vertexv means the number of edges connected to the vertex v denoted by dXv(2)
The numberdXv+dXw−2 is called degree of edgee if e is formed by joining the vertex v and w, denoted by dXe(3)
The setNw=v∈V:vandware nodes ofsome edgee∈E is called the open neighborhood of the vertex w(4)
The setw∪Nw is the closed neighborhood of w denoted by Nw(5)
The number of nonidentical edges that are incident to each vertex in the closed neighbourhood of the vertexv is the ve-degree of vertex v denoted by dXvev [9](6)
Ifvandw are nodes of an edge e, then the order of Nv∪Nw is equal to the ev-degree [9] of edge e, denoted by dXeveFor a graphXV,E, the topological indices under consideration are given below:(i)
Randić index [10]:(1)RX=∑e=1E1dXu×dXv.(ii)
ev-degree Randić index [11]:(2)RevX=∑e=1E1dXeve.(iii)
ve-degree Randić index:(3)RveX=∑e=1E1dXveu×dXvev.(iv)
Reciprocalve-degree Randić index [12]:(4)RRevH=∑e=1EdXeve1/2.Milan Randić, a chemist, introduced the “branching index” in 1975 as a topological indexR for evaluating the degree of branching in the carbon-atom skeleton of saturated hydrocarbons [10]. Moreover, it is also used to model the cavity surface area of different alcohols. In [11], Suleyman Ediz introduced ev−degree Randić (2) index and proved that it gives more accurate correlation than the previous one.First ve-degree Zagreb beta index:(5)MibveX=∑e=1EdXvev+dXvew.Second ve-degree Zagreb index:(6)MiiveX=∑e=1EdXvev×dXvew.Redefined third ve-degree Zagreb index [13]:(7)RZGiiiveH=∑e=1EdXveu×dXvevdXveu+dXvev.The modified ev-degree Zagreb index [12]:(8)∗MevH=∑e=1E1dXeve2.Zagreb indices were first used to model anti-inflammatory agility in different acids [14], and then, M1 and M2 were used to model the clearance of cephalosporins and fraction bound in humans [15].(i)
ve-degree sum-connectivity index:(9)ve−SCX=∑e=1E1dXveu+dXvev.(ii)
ve-degree atom-bond connectivity index:(10)ve−ABCX=∑e=1EdXveu+dXvev−2dXveu×dXvev.Atom-bond connectivity index is used to model heat of formationΔHf in alkanes [16].(iii)
ve-degree geometric arithmetic index:(11)ve−GAX=∑e=1E2dXveu×dXvevdXveu+dXvev.(iv)
Arithmetic-geometric index [17]:(12)ve−AGX=∑e=1EdXveu+dXvev2dXveu×dXvev.In [18], the geometrical-arithmetic index (GA) was introduced and used to model the following properties of chemical compounds:(i)
Acentric factor (AcenFac)(ii)
Boiling point (BP)(iii)
Entropy (S)(iv)
Enthalpy of formation (HFORM)(v)
Enthalpy of vaporization (HVAP)(vi)
Standard enthalpy of vaporization (DHVAP)Recently, some other topological indices (listed below) are introduced on the basis of ev- and ve-degree to model chemical properties written above which give better correlation coefficients of about 0.99088 [19].(i)
ve-degree harmonic index:(13)HveX=∑e=1E2dXveu+dXvev.(ii)
ev-degree inverse index [12]:(14)IDevH=∑e=1E1dXeve.(iii)
ve-degree inverse sum index [20]:(15)ISIveH=∑e=1EdXveu×dXvevdXveu+dXvev.(iv)
F-ev-degree index [12]:(16)FevH=∑e=1EdXeve3.(v)
F-ve-degree index [21]:(17)FveH=∑e=1EdXveu2+dXvev2.(vi)
First hyper-ve-degree index [21]:(18)HM1veH=∑e=1EdXveu+dXvev2.(vii)
Second hyper-ve-degree index [21]:(19)HM2veH=∑e=1EdXveu×dXvev2.For more details, see[22–24].


## 2. Polyphenylene Network

Hydrocarbons, which are organic compounds formulated completely by the atoms of hydrogen and carbon elements, are one of the world’s most prominent sources of energy and are found mostly in petroleum form, natural gas, and crude oil. The majority of hydrocarbons are linear chains, rings, or a mix of the two. Polyphenylene, a two-dimensional (2D) limitless hydrocarbon with the formulaC6H4, that may be formed experimentally, is thermodynamically stable. Furthermore, due to its new structural features, 2D polyphenylene could be used in a variety of industries, such as a barrier for H2 purification. When H2 is produced by the conventional process of steam methane reformation, certain less desirable species are produced, such as CO, CO2, and CH4. As a result, isolating H2 from these species is critical for its storage and use [25]. SRP is a superb option for many demanding applications including semiconductor components, high-performance bearings, bushings, valves, valve seats, and aircraft substructures, due to its remarkable mechanical, chemical, thermal, and electrical qualities. SRP is an ideal contender for light-weight high-performance applications due to its high specific strength. Antistatic coatings made of electrically conductive polyphenylene (p- or n-doped) are used to protect integrated circuits from static charges, humidity, and corrosion [26]. The following are important performance characteristics:Mechanical stiffness and strength are extremely high. High compression strength and resistance to pressure, excellent wear and scratch resistance, and good cold temperature stability (to around−270°C), 155° C is a high glass transition temperature. Before and after processing, there is exceptional dimensional stability, low coefficient of thermal expansion (low thermal shrinkage), excellent acid, and basic resistance. With resistance to solvents and hot steam (but lower than PEEK), processability is good (can be extruded and injection molded).


### 2.1. Mathematical Work and Discussion

In this section, we will compute and discuss all the topological indices mentioned before, for the graph of polyphenylene networkPN, as shown in Figure 1, where, in Figure 2, we show the unit segment of the graph. In the graph, we have vertices joined with single edge, double edges, and triple edges (i.e., dPNv∈1,2,3). In the graph of polyphenylene network PNV,E, we observe that there are 30m2+8m−4 vertices and 34m2+m+3 edges. More information about the graph is given in Table 1.(20)ave−SCPN=33+522+710+415m2+m23−122+123+110+215,bve−ABCPN=1425+10+182/7+26/7m2+2/7m+225−2/5+2/7+13/14.Figure 1
Molecular graph of polyphenylene network.Figure 2
Unit of the graph of polyphenylene network.Table 1
Edge partition table of polyphenylene network.
dPNu,dPNvdPNedPNveu,dPNvevdPNeveFrequency(1, 3)2(3, 5)45m2−1(2, 2)2(5, 5)47m2+1(2, 3)3(5, 7)518m2+m+1(3, 3)4(7, 8)64m2+2Theorem 1.
LetPNV,Ebe the graph of polyphenylene network; then,(21)aRPN=16103+186+29m2+m6+16−23+6+7,bRevPN=22/3+185+6m2+m5+2/3+15,cRvePN=5/3+2/7+75+1835m2+m35+15+114−115+135,dRRevPN=295+26+12m2+5m+5+26.Proof.
To prove (a), we expand equation (1) and substitute values from Table 1, which gives(22)RPN=4m2+23×3+5m2−11×3+18m2+m+12×3+7m2+12×2=134m2+2+127m2+1+5m2−13+18m2+m+16=16103+186+29m2+6m−23+6+7=16103+186+29m2+m6+16−23+6+7.
To prove (b), we expand (2) and substitute values from Table 1, which gives(23)RevPN=4m2+26+5m2−14+18m2+m+15+7m2+14=4m2+26+125m2−1+127m2+1+18m2+m+15=22/3+185+6m2+m5+2/3+15.
To prove (c), we expand (3) and substitute values from Table 1, which gives(24)RvePN=4m2+27×8+5m2−13×5+18m2+m+15×7+7m2+15×5=2m2+114+157m2+1+5m2−115+18m2+m+135=5/32/7+75+1835m2+m35+15+114−115+135.
To prove (d), we expand (4) and substitute values from Table 1, which gives(25)RRevPN=64m2+2+45m2−1+518m2+m+1+47m2+1=25m2−1+27m2+1+518m2+m+1+64m2+2=295+26+12m2+5m+5+26.Theorem 2.
LetPNV,Ebe the graph of polyphenylene network; then,(26)aMibvePN=386m2+12m+44,bMiivePN=1104m2+35m+157,cRZGiiivePN=10223+42m+1327m2,d∗MevPN=19001423m2+36m+86.Proof.
To prove (a), expand equation (5) and substitute values from Table 1, which gives(27)MibvePN=7+84m2+2+3+55m2−1+5+718m2+m+1=+5+57m2+1154m2+2+85m2−1+107m2+1+1218m2+m+1=386m2+12m+44.
To prove (b), expand (6) and substitute values from Table 1, which gives(28)MiivePN=7×84m2+2+3×55m2−1+5×718m2+m+1+5×57m2+1=564m2+2+155m2−1+257m2+1+3518m2+m+1=1104m2+35m+157.
To prove (c), expand (7) and substitute values from Table 1, which gives(29)RZGiiivePN=5m2−13×53+5+7m2+15×55+5+18m2+m+15×75+7+4m2+27×87+8=8402+4m2+120−1+5m2+2501+7m2+4201+m+18m2=10223+42m+1327m2.
To prove (d), expand (8) and substitute values from Table 1, which gives(30)∗MevPN=5m2−1×142+7m2+1×142+18m2+m+1×152+4m2+2×162=1364m2+2+1165m2−1+1167m2+1+12518m2+m+1=19001423m2+36m+86.Theorem 3.
LetPNV,E be the graph of polyphenylene network; then,Proof.
To prove (a), expanding equation (9) and substituting values from Table 1, we find(31)ve−SCPN=4m2+27+8+5m2−13+5+18m2+m+15+7+7m2+15+5=4m2+215+5m2−122+18m2+m+123+7m2+110=33+522+710+415m2+m23−122+123+110+215.
To prove (b), expanding (10) and substituting values from Table 1, we find(32)ve−ABCPN=7+8−2/7×84m2+2+3+5−2/3×55m2−1+5+7−2/5×718m2+m+1+5+5−2/5×57m2+1=2527m2+1+2/55m2−1+2/718m2+m+1+13/142m2+1=1425+10+182/7+26/7m2+2/7m+225−2/5+2/7+13/14$.Theorem 4.
LetPNV,Ebe the graph of polyphenylene network, then(33)ave−GAPN=161415+5154+335+7m2+35m6+81415+356−154+1,bve−AGPN=453+1514+10835+7m2+6m35+15214−415+635+1.Proof.
To prove (a), expand equation (11) and substitute values from Table 1, which gives(34)ve−GAPN=4m2+227×87+8+5m2−123×53+5+18m2+m+125×75+7+7m2+125×55+5=7m2+415144m2+2+14155m2−1+163518m2+m+1+1=161415+5154+335+7m2+35m6+81415+356−154+1.
To prove (b), expand (12) and substitute values from Table 1, which gives(35)AGvePN=5m2−13+523×5+7m2+15+525×5+18m2+m+17+527×5+4m2+27+827×8=1+7m2+151+2m2214+4−1+5m215+61+m+18m235=453+1514+10835+7m2+6m35+15214−415+635+1.Theorem 5.
LetPNV,E be the graph of polyphenylene network; then,(36)HvePN=160371m2+10m+23.Proof.
To prove, expand equation (13) and substitute values from Table 1, which gives(37)HvePN=24m2+27+8+25m2−13+5+218m2+m+15+7+27m2+15+5=2154m2+2+145m2−1+157m2+1+1618m2+m+1=160371m2+10m+23.Theorem 6.
LetPNV,E be the graph of polyphenylene network; then,(38)aIDevPN=115109m2+3m+8,bISIvePN=11201321+350m+11317m2.Proof.
To prove (a), expand equation (14) and substitute values from Table 1, which gives(39)IDevPN=5m2−1×14+7m2+1×14+18m2+m+1×15+4m2+2×16=115109m2+3m+8.
To prove (b), expand (15) and substitute values from Table 1, which gives(40)ISIvePN=5m2−13×53+5+7m2+15×55+5+18m2+m+17×57+5+4m2+27×87+8=56152+4m2+158−1+5m2+521+7m2+35121+m+18m2=11201321+350m+11317m2.Theorem 7.
LetPNV,E be the graph of polyphenylene network; then,(41)aFevPN=3882m2+125m+557,bFvePN=316+74m+2304m2.Proof.
To prove (a), expand equation (16) and substitute values from Table 1, which gives(42)FevPN=435m2−1+437m2+1+5318m2+m+1+634m2+2=2164m2+2+645m2−1+647m2+1+12518m2+m+1=3882m2+125m+557.
To prove (b), expand (17) and substitute values from Table 1, which gives(43)FvePN=5m2−132+52+7m2+152+52+18m2+m+152+72+4m2+272+82=1132+4m2+34−1+5m2+501+7m2+741+m+18m2=316+74m+2304m2.Theorem 8.
LetPNV,E be the graph of polyphenylene network; then,(44)aHM1vePN=6105+24m+752m2,bHM2vePN=7897+1225m+40094m2.Proof.
To prove (a), expand equation (18) and substitute values from Table 1, which gives(45)HM1vePN=5m2−13+52+7m2+15+52+18m2+m+15+72+4m2+27+82=2252+4m2+64−1+5m2+1001+7m2+1441+m+18m2=6105+24m+752m2.
To prove (b), expand (19) and substitute values from Table 1, which gives(46)HM2vePN=5m2−13×52+7m2+15×52+18m2+m+15×72+4m2+27×82=31362+4m2+225−1+5m2+6251+7m2+12251+m+18m2=7897+1225m+40094m2.


### 2.2. Numerical and Graphical Discussion

In Theorem1, we stated and then computed Randić indices of the graph of polyphenylene network (Figure 1) for m frameworks or units. By the resulting equations, we determined exact numerical values of Randić indices, for some m=1,2,…,10, and listed in Table 2 and plotted them in Figure 3 by which we can observe that all the resulting equations show parabolic behavior. For continuously increasing values of m, all the indices go to infinity. RRevPN increases faster than other indices which are in order RvePN<RPN<RevPN.Table 2
The numerical representation of Randic-type indices.
mRPNRevPNRvePNRRevPN116.474 417.393 86.815 283.418 3262.088 364.889 525.788 4307.7963137.839143.75157.297 8680.2684243.727253.978101.3431200.835379.753395.571157.9251869.496545.915568.529227.0432686.257742.214772.853308.6973651.18968.6511008.54402.8874764.0491225.221275.6509.6136025.08101511.941574.02628.8767434.21Figure 3
The graphical representation of Randic-type indices.Table3 and Figure 4 are derived from Theorem 2 which show the numerical values and graphical pattern of the Zagreb indices of polyphenylene network. The curves are parabolic, and the parabola for RZGiiivePN has the shortest length of lactusrectum; i.e., it runs much faster than the others.Table 3
The numerical representation of Zagreb-type indices.
mMibvePNMiivePNRZGiiivePN∗MvePN1442129615 9201.716 6721612464356 1506.53355410 198122 92014.445 64626817 961216 23025.553 35975427 932336 08039.823 3614 01240 111482 47057.255 6719 04254 498655 40077.85824 84471 093854 870101.607931 41889 8961080 880128.5261038 764110 9071333 430158.607Figure 4
The graphical representation of Zagreb-type indices.In Table4 and Figure 5, we exhibited the numerical measures and graphical style of sum-connectivity and atom-bond connectivity indices from Theorem 3. The curves are parabolic, and the separation between them becomes greater and greater for increasing values of m.ve−ABCPN earns larger numerical value than ve−SCPN for all values of m.Table 4
The numerical representation of ve-ABC and ve-SC indices.
mve - ABC(PN)ve - SC(PN)120.636 611.266 7277.183 342.186 33171.07293.526 64302.301165.2875470.872257.4696676.785370.0717920.039503.09481200.63656.53791518.57830.401101873.851024.69Figure 5
The graphical representation of ve-ABC and ve-SC indices.Form units of polyphenylene, ve−GAPN and ve−AGPN were given in Theorem 4. By substituting m=1,…,10, Table 5 is constructed. In Figure 6, the curves of both the indices are shown from where we can observe that the span between them is growing gradually.Table 5
The numerical representation of the geometric arithmetic index.
mve - GA(PN)ve - AG(PN)137.579 938.428 32139.308142.7273308.196315.8824544.246557.8945847.458868.76261217.831248.4971655.361697.0782160.062214.5192731.912800.8103370.933455.95Figure 6
The graphical representation of the geometric arithmetic index.In Figure7, the outcomes of Theorem 5 are shown. The numerical values are increasing continuously with the increase of frameworks m.Figure 7
The graphical representation of ve-degree harmonic indices.Table6 and Figure 8 are established for Theorem 6 in which we can see that, for all m,IDevPN<ISIvePN, and both the curves are parabolic in manner. The span between the curves is growing very high.Table 6
The numerical representation of inverse indices.
mIDevPNISIvePN18108.233230394.075366.533 3868.5334117.61531.615183.22383.36263.3333423.6173584652.538467.26070.079590.9337676.2310729.29471.01Figure 8
The graphical representation of inverse indices.Numerical values of the indices discussed in Theorem7 are given in Table 7, where the graph of both the indices is shown in Figure 9. Again, both the curve are parabolic. The index FevPN acquires higher values than the index FvePN all the time, for the graph of polyphenylene network.Table 7
The numerical representation of F-ev-degree index.
mFevPNFvePN145642694216 3359680335 87021 274463 16937 476598 23258 2866141 05983 7047191 650113 7308250 005148 3649316 124187 60610390 007231 456Figure 9
The graphical representation of F-ev-degree index.Following are the conclusions of final Theorem8. The numerical representation is shown in Table 8, and graphical representation is shown in Figure 10. Once again both the curves are parabolic. The separation between them is growing continuously. The numerical values of HM1vePN is always less than the numerical values of HM2vePN.Table 8
The numerical representation of hyper-ve-degree index.
mHM1vePNHM2vePN1528649 216218 966170 723341 670372 418473 398654 3015114 1501.01637×1066163 9261.45863×1067222 7261.98108×1068290 5502.58371×1069367 3983.26654×10610453 2704.02955×106Figure 10
The graphical representation of hyper-ve-degree index.


## 2.1. Mathematical Work and Discussion

In this section, we will compute and discuss all the topological indices mentioned before, for the graph of polyphenylene networkPN, as shown in Figure 1, where, in Figure 2, we show the unit segment of the graph. In the graph, we have vertices joined with single edge, double edges, and triple edges (i.e., dPNv∈1,2,3). In the graph of polyphenylene network PNV,E, we observe that there are 30m2+8m−4 vertices and 34m2+m+3 edges. More information about the graph is given in Table 1.(20)ave−SCPN=33+522+710+415m2+m23−122+123+110+215,bve−ABCPN=1425+10+182/7+26/7m2+2/7m+225−2/5+2/7+13/14.Figure 1
Molecular graph of polyphenylene network.Figure 2
Unit of the graph of polyphenylene network.Table 1
Edge partition table of polyphenylene network.
dPNu,dPNvdPNedPNveu,dPNvevdPNeveFrequency(1, 3)2(3, 5)45m2−1(2, 2)2(5, 5)47m2+1(2, 3)3(5, 7)518m2+m+1(3, 3)4(7, 8)64m2+2Theorem 1.
LetPNV,Ebe the graph of polyphenylene network; then,(21)aRPN=16103+186+29m2+m6+16−23+6+7,bRevPN=22/3+185+6m2+m5+2/3+15,cRvePN=5/3+2/7+75+1835m2+m35+15+114−115+135,dRRevPN=295+26+12m2+5m+5+26.Proof.
To prove (a), we expand equation (1) and substitute values from Table 1, which gives(22)RPN=4m2+23×3+5m2−11×3+18m2+m+12×3+7m2+12×2=134m2+2+127m2+1+5m2−13+18m2+m+16=16103+186+29m2+6m−23+6+7=16103+186+29m2+m6+16−23+6+7.
To prove (b), we expand (2) and substitute values from Table 1, which gives(23)RevPN=4m2+26+5m2−14+18m2+m+15+7m2+14=4m2+26+125m2−1+127m2+1+18m2+m+15=22/3+185+6m2+m5+2/3+15.
To prove (c), we expand (3) and substitute values from Table 1, which gives(24)RvePN=4m2+27×8+5m2−13×5+18m2+m+15×7+7m2+15×5=2m2+114+157m2+1+5m2−115+18m2+m+135=5/32/7+75+1835m2+m35+15+114−115+135.
To prove (d), we expand (4) and substitute values from Table 1, which gives(25)RRevPN=64m2+2+45m2−1+518m2+m+1+47m2+1=25m2−1+27m2+1+518m2+m+1+64m2+2=295+26+12m2+5m+5+26.Theorem 2.
LetPNV,Ebe the graph of polyphenylene network; then,(26)aMibvePN=386m2+12m+44,bMiivePN=1104m2+35m+157,cRZGiiivePN=10223+42m+1327m2,d∗MevPN=19001423m2+36m+86.Proof.
To prove (a), expand equation (5) and substitute values from Table 1, which gives(27)MibvePN=7+84m2+2+3+55m2−1+5+718m2+m+1=+5+57m2+1154m2+2+85m2−1+107m2+1+1218m2+m+1=386m2+12m+44.
To prove (b), expand (6) and substitute values from Table 1, which gives(28)MiivePN=7×84m2+2+3×55m2−1+5×718m2+m+1+5×57m2+1=564m2+2+155m2−1+257m2+1+3518m2+m+1=1104m2+35m+157.
To prove (c), expand (7) and substitute values from Table 1, which gives(29)RZGiiivePN=5m2−13×53+5+7m2+15×55+5+18m2+m+15×75+7+4m2+27×87+8=8402+4m2+120−1+5m2+2501+7m2+4201+m+18m2=10223+42m+1327m2.
To prove (d), expand (8) and substitute values from Table 1, which gives(30)∗MevPN=5m2−1×142+7m2+1×142+18m2+m+1×152+4m2+2×162=1364m2+2+1165m2−1+1167m2+1+12518m2+m+1=19001423m2+36m+86.Theorem 3.
LetPNV,E be the graph of polyphenylene network; then,Proof.
To prove (a), expanding equation (9) and substituting values from Table 1, we find(31)ve−SCPN=4m2+27+8+5m2−13+5+18m2+m+15+7+7m2+15+5=4m2+215+5m2−122+18m2+m+123+7m2+110=33+522+710+415m2+m23−122+123+110+215.
To prove (b), expanding (10) and substituting values from Table 1, we find(32)ve−ABCPN=7+8−2/7×84m2+2+3+5−2/3×55m2−1+5+7−2/5×718m2+m+1+5+5−2/5×57m2+1=2527m2+1+2/55m2−1+2/718m2+m+1+13/142m2+1=1425+10+182/7+26/7m2+2/7m+225−2/5+2/7+13/14$.Theorem 4.
LetPNV,Ebe the graph of polyphenylene network, then(33)ave−GAPN=161415+5154+335+7m2+35m6+81415+356−154+1,bve−AGPN=453+1514+10835+7m2+6m35+15214−415+635+1.Proof.
To prove (a), expand equation (11) and substitute values from Table 1, which gives(34)ve−GAPN=4m2+227×87+8+5m2−123×53+5+18m2+m+125×75+7+7m2+125×55+5=7m2+415144m2+2+14155m2−1+163518m2+m+1+1=161415+5154+335+7m2+35m6+81415+356−154+1.
To prove (b), expand (12) and substitute values from Table 1, which gives(35)AGvePN=5m2−13+523×5+7m2+15+525×5+18m2+m+17+527×5+4m2+27+827×8=1+7m2+151+2m2214+4−1+5m215+61+m+18m235=453+1514+10835+7m2+6m35+15214−415+635+1.Theorem 5.
LetPNV,E be the graph of polyphenylene network; then,(36)HvePN=160371m2+10m+23.Proof.
To prove, expand equation (13) and substitute values from Table 1, which gives(37)HvePN=24m2+27+8+25m2−13+5+218m2+m+15+7+27m2+15+5=2154m2+2+145m2−1+157m2+1+1618m2+m+1=160371m2+10m+23.Theorem 6.
LetPNV,E be the graph of polyphenylene network; then,(38)aIDevPN=115109m2+3m+8,bISIvePN=11201321+350m+11317m2.Proof.
To prove (a), expand equation (14) and substitute values from Table 1, which gives(39)IDevPN=5m2−1×14+7m2+1×14+18m2+m+1×15+4m2+2×16=115109m2+3m+8.
To prove (b), expand (15) and substitute values from Table 1, which gives(40)ISIvePN=5m2−13×53+5+7m2+15×55+5+18m2+m+17×57+5+4m2+27×87+8=56152+4m2+158−1+5m2+521+7m2+35121+m+18m2=11201321+350m+11317m2.Theorem 7.
LetPNV,E be the graph of polyphenylene network; then,(41)aFevPN=3882m2+125m+557,bFvePN=316+74m+2304m2.Proof.
To prove (a), expand equation (16) and substitute values from Table 1, which gives(42)FevPN=435m2−1+437m2+1+5318m2+m+1+634m2+2=2164m2+2+645m2−1+647m2+1+12518m2+m+1=3882m2+125m+557.
To prove (b), expand (17) and substitute values from Table 1, which gives(43)FvePN=5m2−132+52+7m2+152+52+18m2+m+152+72+4m2+272+82=1132+4m2+34−1+5m2+501+7m2+741+m+18m2=316+74m+2304m2.Theorem 8.
LetPNV,E be the graph of polyphenylene network; then,(44)aHM1vePN=6105+24m+752m2,bHM2vePN=7897+1225m+40094m2.Proof.
To prove (a), expand equation (18) and substitute values from Table 1, which gives(45)HM1vePN=5m2−13+52+7m2+15+52+18m2+m+15+72+4m2+27+82=2252+4m2+64−1+5m2+1001+7m2+1441+m+18m2=6105+24m+752m2.
To prove (b), expand (19) and substitute values from Table 1, which gives(46)HM2vePN=5m2−13×52+7m2+15×52+18m2+m+15×72+4m2+27×82=31362+4m2+225−1+5m2+6251+7m2+12251+m+18m2=7897+1225m+40094m2.


## 2.2. Numerical and Graphical Discussion

In Theorem1, we stated and then computed Randić indices of the graph of polyphenylene network (Figure 1) for m frameworks or units. By the resulting equations, we determined exact numerical values of Randić indices, for some m=1,2,…,10, and listed in Table 2 and plotted them in Figure 3 by which we can observe that all the resulting equations show parabolic behavior. For continuously increasing values of m, all the indices go to infinity. RRevPN increases faster than other indices which are in order RvePN<RPN<RevPN.Table 2
The numerical representation of Randic-type indices.
mRPNRevPNRvePNRRevPN116.474 417.393 86.815 283.418 3262.088 364.889 525.788 4307.7963137.839143.75157.297 8680.2684243.727253.978101.3431200.835379.753395.571157.9251869.496545.915568.529227.0432686.257742.214772.853308.6973651.18968.6511008.54402.8874764.0491225.221275.6509.6136025.08101511.941574.02628.8767434.21Figure 3
The graphical representation of Randic-type indices.Table3 and Figure 4 are derived from Theorem 2 which show the numerical values and graphical pattern of the Zagreb indices of polyphenylene network. The curves are parabolic, and the parabola for RZGiiivePN has the shortest length of lactusrectum; i.e., it runs much faster than the others.Table 3
The numerical representation of Zagreb-type indices.
mMibvePNMiivePNRZGiiivePN∗MvePN1442129615 9201.716 6721612464356 1506.53355410 198122 92014.445 64626817 961216 23025.553 35975427 932336 08039.823 3614 01240 111482 47057.255 6719 04254 498655 40077.85824 84471 093854 870101.607931 41889 8961080 880128.5261038 764110 9071333 430158.607Figure 4
The graphical representation of Zagreb-type indices.In Table4 and Figure 5, we exhibited the numerical measures and graphical style of sum-connectivity and atom-bond connectivity indices from Theorem 3. The curves are parabolic, and the separation between them becomes greater and greater for increasing values of m.ve−ABCPN earns larger numerical value than ve−SCPN for all values of m.Table 4
The numerical representation of ve-ABC and ve-SC indices.
mve - ABC(PN)ve - SC(PN)120.636 611.266 7277.183 342.186 33171.07293.526 64302.301165.2875470.872257.4696676.785370.0717920.039503.09481200.63656.53791518.57830.401101873.851024.69Figure 5
The graphical representation of ve-ABC and ve-SC indices.Form units of polyphenylene, ve−GAPN and ve−AGPN were given in Theorem 4. By substituting m=1,…,10, Table 5 is constructed. In Figure 6, the curves of both the indices are shown from where we can observe that the span between them is growing gradually.Table 5
The numerical representation of the geometric arithmetic index.
mve - GA(PN)ve - AG(PN)137.579 938.428 32139.308142.7273308.196315.8824544.246557.8945847.458868.76261217.831248.4971655.361697.0782160.062214.5192731.912800.8103370.933455.95Figure 6
The graphical representation of the geometric arithmetic index.In Figure7, the outcomes of Theorem 5 are shown. The numerical values are increasing continuously with the increase of frameworks m.Figure 7
The graphical representation of ve-degree harmonic indices.Table6 and Figure 8 are established for Theorem 6 in which we can see that, for all m,IDevPN<ISIvePN, and both the curves are parabolic in manner. The span between the curves is growing very high.Table 6
The numerical representation of inverse indices.
mIDevPNISIvePN18108.233230394.075366.533 3868.5334117.61531.615183.22383.36263.3333423.6173584652.538467.26070.079590.9337676.2310729.29471.01Figure 8
The graphical representation of inverse indices.Numerical values of the indices discussed in Theorem7 are given in Table 7, where the graph of both the indices is shown in Figure 9. Again, both the curve are parabolic. The index FevPN acquires higher values than the index FvePN all the time, for the graph of polyphenylene network.Table 7
The numerical representation of F-ev-degree index.
mFevPNFvePN145642694216 3359680335 87021 274463 16937 476598 23258 2866141 05983 7047191 650113 7308250 005148 3649316 124187 60610390 007231 456Figure 9
The graphical representation of F-ev-degree index.Following are the conclusions of final Theorem8. The numerical representation is shown in Table 8, and graphical representation is shown in Figure 10. Once again both the curves are parabolic. The separation between them is growing continuously. The numerical values of HM1vePN is always less than the numerical values of HM2vePN.Table 8
The numerical representation of hyper-ve-degree index.
mHM1vePNHM2vePN1528649 216218 966170 723341 670372 418473 398654 3015114 1501.01637×1066163 9261.45863×1067222 7261.98108×1068290 5502.58371×1069367 3983.26654×10610453 2704.02955×106Figure 10
The graphical representation of hyper-ve-degree index.


## 3. Conclusion

In this study, we calculated 19 topological indices discussed in Section1 for the graph of polyphenylene network. Each index represents a unique chemical or physical property which is also mentioned in section 1. We also mentioned many qualities of polyphenylene which tell that it is an admirable chemical compound to make new substances and to improve dispositions of the present one. Therefore, the work of this paper will provide assistance in studying the properties of polyphenylene and its derivatives.



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