# Study of HCP (Hexagonal Close-Packed) Crystal Structure Lattice through Topological Descriptors

Authors: Guoping Zhang; Saadia Saeed; Adnan Aslam; Salma Kanwal; Nazeran Idrees; Tahira Sumbal Shaikh
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

Chemical graph theory is a multidisciplinary field where the structure of the molecule is analyzed as a graphical structure. Chemical descriptors are one of the most important ideas employed in chemical graph theory; this is to associate a numerical value with a graph structure that often has correlation with corresponding chemical properties. In this paper, we investigate another very important closed-packed usual crystal structure defined asHCP (Hexagonal Close-Packed) crystal structure and its lattice formed by arranging its unit cells in a dimension for topological descriptors based on a neighborhood degree, reverse degree, and degree. Furthermore, we classify which descriptor is more dominating.


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## Body

## 1. Introduction

Chemical graph theory plays a considerate role to investigate a wide range of inorganic and organic chemical structures by studying them through graphical representation. A topological descriptor is a mathematical entity based on certain topological features of chemical structure which correlates with corresponding chemical properties [1–7]. In this research work, we study the most symmetrical and complex close-packed structure defined as HCPn using two-dimensional topological descriptors. In graph theory [8], degree of a vertex s in a corresponding graph is the number of edges incident with that vertex defined as ds. Reverse degree of a vertex s is defined as ℝs=ΔG+1−ds, where ΔG is the maximum degree of a vertex in a graph G. Neighborhood degree of a vertex s in G is defined as δs=∑t∈Nsdt, where NS represents the neighborhood degree of a vertex s.Hexagonal Close PackingHCP consists of alternating layers of spheres of atoms (vertices) arranged in a hexagon, with one additional atom (vertex) at the center as shown in Figure 1. Another layer of atoms is sandwiched between these two hexagonal layers which are triangular (three atoms form a triangle by three edges), and the atoms (vertices) of this layer fill the tetrahedral holes created by the top and bottom [9]. The edge set EG of HCP contains the edges that connect the atoms that are nearest to each other by an edge. In this context, we represent edges by both a filled and a dotted line to clarify the bonding between two atoms in a 3D HCPn. The middle layer atoms also share a bond with both hexagonal layers by following a symmetrical pattern.Figure 1
HCPn; hexagonal close packing lattice formed by n unit cells.Elements that form anHCP crystal structure are zirconium, ruthenium, hafnium, and many more. It is the most strong and brittle structure that is highly found in metals that do not have a very smooth symmetry as in cubic structures, but it contains a vast and stronger metallic property than a usual hexagonal crystal structure.


## 2. Preliminaries

We give a brief view of some well-known TIs for which we compute closed formulas corresponding to each crystal structure.Every index is defined as(1)TIG=∑edgesfH,K.In Table1, we give a brief view for each descriptor in terms of its function defining in above equation as fH,K.HCPn is a very densely symmetric crystal structure with a complex bonding between its atoms. As it is not a tough thing to study any planar chem structure via descriptors but, here our goal is to perform evaluation for not a planar but a 3-dimensional structure with a complex symmetry in comparison to cubic structure. We regard the study for the work done to date for complex and 3D crystals, including diamond crystal structures [23], FCC [24], and BCC [25] crystal unit cells that have been investigated by utilizing the definitions from chemical graph theory [5], and allows to study organic chemistry, cordially here, metals and minerals attaining crystal unit cells.Table 1
Topological indices and their corresponding functions.
FoundersNameDefining H and KTI=fH,KGutman [10]FirstZagrebH=dsandK=dtM1G=H+KSecond ZagrebM2G=H×KBollobás et al. and Amić et al. [11, 12]General Randić indexRαG=H×KαEstrada et al. [13]Atom bond connectivity indexABCG=H+K−2/H×KVukičević et al. [14]Geometric arithmetic indexGAG=2H×K/H+KChen et al. [15]Exponential reduced Zagreb indexeRM2G=eH−1K−1Zhao et al. [16]General Reverse Randić index,whereα∈RH=ℜsandK=ℜtRαG=H×KαEstrada et al. [13]Reverse Atom Bond Connectivity indexℝABCG=H+K−2/H×KVukičević et al. [13]Reverse geometric arithmetic indexℝGAG=2H×K/H+KShirdel et al. [17]Reverse hyper Zagreb indexℝHMG=H+K2Ghorbani et al. [18]Neighborhood versio of first Zagreb indexH=δsandK=δtM1∗G=H+KMonadal [19]Neighborhood second Zagreb indexM2∗G=H×KNeighborhood Forgotten indexFN∗G=H2+K2Siddiqui et al. [20]Neighborhood second modified Zagreb indexM2nmG=1/H×KPal et al. [21]ThirdNDeND3=HKH+KFifthNDeND5=H/K+K/HNeighborhood Harmonic indexNHG=2/H+KNeighborhood Inverse sum indexNIG=H×K/H+KHosamani et al. [22]The Sanskruti indexSG=H×K/H+K−23


## 3. Formulation

In this section, we consider that a graphG is HCPn defined as hexagonal close-packed crystal structure lattice consisting of n unit cells arranged in one dimension. VHCPn is defined as vertex set of hexagonal close-packed lattice, and edge set is defined as EHCPn where for n ordered lattice VHCPn=10n+7, EHCPn=40n+12.


### 3.1. Degree Based

We utilize Table2 to compute closed forms for the defined descriptors dependent on degree.(1)
First Zagreb index ofHCPn is defined as follows:(2)M1G=∑s∈VGds+dt=1210+1215+1213+3n16+618+12n−1814+12n−1215+6n−621+224+n−228+6n−622+1212=634n−10.(2)
Second Zagreb index ofHCPn is defined as follows:(3)M2G=∑st∈EGdsdt=1225+1250+1240+3n64+680+12n−1849+12n−1256+6n−698+2140+n−2196+6n−6112+1235=2908n−646.(3)
General Randić index ofHCPn is defined as follows:(4)RαG=∑st∈EGdsdtα=12dsdtα+12dsdtα+12dsdtα+3ndsdtα+6dsdtα+12n−18dsdtα+12n−12dsdtα+6n−6dsdtα+2dsdtα+n−2dsdtα+6n−6dsdtα+12dsdtα.(5)R1G=∑st∈EGdsdt1=12251+12501+12401+3n641+6801+12n−18491+12n−12561+6n−6981+21401+n−21961+6n−61121+12351=M2G.(6)R−1G=∑st∈EGdsdt−1=1225−1+1250−1+1240−1+3n64−1+680−1+12n−1849−1+12n−1256−1+6n−698−1+2140−1+n−2196−1+6n−6112−1+1235−1=0.7455102041+0.5790816327n.(7)R12G=∑st∈EGdsdt12=122512+125012+124012+3n6412+68012+12n−184912+12n−125612+6n−69812+214012+n−219612+6n−611212+123512=2.375607208+334.6947784n.(8)R−12G=∑st∈EGdsdt−12=1225−12+1250−12+1240−12+3n64−12+680−12+12n−1849−12+12n−1256−12+6n−698−12+2140−12+n−2196−12+6n−6112−12+1235−12=3.371752924+4.937319973n.(3a)
Whenα=1,(3b)
Whenα=−1,(3c)
Whenα=1/2,(3d)
Whenα=−1/2,(4)
Atom bond connectivity index ofHCPn is defined as follows:(9)ABCG=∑st∈EGds+dt−2dsdt=12825+121350+121140+3n732+615+12n−181249+12n−121356+6n−61998+21170+n−21398+6n−6528+1227=8.495057575+18.66489626n.(5)
The geometric arithmetic index ofHCPn is defined as follows:(10)GAG=∑st∈EG2dsdtds+dt=1222510+1225015+1224013+3n26416+628018+12n−1824914+12n−1225615+6n−629821+2214024+n−2219628+6n−6211222+1223512=39.4027062n+11.35413853.(6)
Exponential Reduced Zagreb index ofHCPn is defined as follows:(11)eRM2G=∑st∈EGeds−1dt−1=12e16+12e36+12e28+3ne49+6e63+12n−18e36+12n−12e42+6n−6e78+2e117+n−2e169+6n−6e91+12e24=2.487524928×1073n−4.975049857×1073.Table 2
Edge partition ofHCPn based on degree of VHCPn
Edge representationds,dtCardinalityofds,dtE15,512E25,1012E35,812E48,83nE58,106E67,712n−18E77,812n−12E87,146n−6E910,142E1014,14n−2E118,146n−6E125,712


### 3.2. Reverse Degree Based

We utilize Table3 to compute closed forms for the defined descriptors dependent on reverse degree of a vertex set of HCPn. where maximum degree for HCPn=ΔHCPn=14, which we use to compute reverse degree for the entire vertex set of lattice.(1)
Reverse Randić index ofHCPn is as follows where α=±1,±1/2:(1a)
Whenα=1,(12)ℝR1G=∑st∈EGℝsℝt1=∑st∈E1ℝsℝt+∑st∈E2ℝsℝt+⋯+∑st∈E12ℝsℝt=1210×10+1210×5+1210×7+3n7×7+67×5+12n−188×8+12n−128×7+6n−68×1+25×1+n−21×1+6n−67×1+1210×8=1904+1678n.(1b)
Whenα=−1,(13)ℝR−1G=∑st∈EGℝsℝt−1=∑st∈E1ℝsℝt−1+∑st∈E2ℝsℝt−1+⋯+∑st∈E12ℝsℝt−1=1210×10−1+1210×5−1+1210×7−1+3n7×7−1+67×5−1+12n−188×8−1+12n−128×7−1+6n−68×1−1+25×1−1+n−21×1−1+6n−67×1−1+1210×8−1=2407n784−159595600.(1c)
Whenα=1/2,(14)ℝR12G=∑st∈EGℝsℝt12=∑st∈E1ℝsℜt12+∑st∈E2ℝsℝt12+⋯+∑st∈E12ℝsℝt12=1210×1012+1210×512+1210×712+3n7×712+67×512+12n−188×812+12n−128×712+6n−68×112+25×112+n−21×112+6n−67×112+1210×812=240.6448479n+183.9070466.(1d)
Whenα=−1/2,(15)ℝR−12G=∑st∈EGℝsℝt−12=∑st∈E1ℝsℝt−12+∑st∈E2ℝsℝt−12+⋯+∑st∈E12ℝsℝt−12=1210×10−12+1210×5−12+1210×7−12+3n7×7−12+67×5−12+12n−188×8−12+12n−128×7−12+6n−68×1−12+25×1−12+n−21×1−12+6n−67×1−12+1210×8−12=8.921246062n−3.503847478.(2)
Reverse atom bond connectivity index ofHCPn is defined as follows:(16)ℝABCG=∑st∈E1ℝs+ℝt−2ℝsℝt=∑st∈E2ℝs+ℝt−2ℝsℝt+∑st∈E3ℝs+ℝt−2ℝsℝt+⋯+∑st∈E12ℝs+ℝt−2ℝsℝt=1210+10−210×10+1210+5−210×5+1210+7−210×7+3n7+7−27×7+67+5−27×5+12n−188+8−28×8+12n−128+7−28×7+6n−68+1−28×1+25+1−25×1+n−21+1−21×1+6n−67+1−27×2+1210+8−2108=24.04625241n+1.759584801.(3)
Reverse geometric arithmetic index ofHCPn is defined as follows:(17)ℝGAG=∑st∈EG2ℝsℝtℝs+ℝt=∑st∈E12ℝsℝtℝs+ℝt+∑st∈E22ℝsℝtℝs+ℝt+∑st∈E32ℝsℝtℝs+ℝt+⋯+∑st∈E122ℝsℝtℝs+ℝt=12210×1010+10+12210×510+5+12210×710+7+3n27×77+7+627×57+5+12n−1828×88+8+12n−1228×78+7+6n−628×18+1+225×15+1+n−221×11+1+6n−627×17+1+12210×810+8=35.71316677n+14.74470034.(4)
Reverse hyper Zagreb index ofHCPn is defined as follows:(18)ℝHMG=∑st∈EGℝs+ℝt2=∑st∈E1ℝs+ℝt2+∑st∈E2ℝs+ℝt2+∑st∈E3ℝs+ℝt2+⋯+∑st∈E12ℝs+ℝt2=1210+102+1210+52+1210+72+3n7+72+67+52+12n−188+82+12n−128+72+6n−68+12+25+12+n−21+12+6n−67+12+1210+82=7234n+7606.Table 3
Edge partition ofHCPn based on reverse degree of VHCPn
Edge representationℝs,ℝtCardinality ofℝs,ℝtE110,1012E210,512E310,712E47,73nE57,56E68,812n−18E78,712n−12E88,16n−6E95,12E101,1n−2E117,16n−6E1210,812


### 3.3. Neighborhood Degree Based

We utilize Table4 to evaluate closed formulas for the neighborhood degree-based descriptors representing topological properties.(1)
Neighborhood version of first Zagreb index ofHCPn is defined as follows:(19)M1∗G=∑st∈EGδs+δt=∑st∈E1δs+δt+∑st∈E2δs+δt+∑st∈E3δs+δt+⋯+∑st∈E21δs+δt=1270+12103+1299+1291+2182+12112+12120+6178+12170+12128+6186+2232+6128+3n−6144+12114+12n−42116+12n−36130+6n−18176+6n−18190+6132+n−4236=4792+5816n.(2)
The neighborhood secondZagreb index of HCPn is defined as follows:(20)M2∗G=∑st∈EGδsδt=∑st∈E1δsδt+∑st∈E2δsδt+⋯+∑st∈E21δsδt=121225+122380122240+12196027752+123136+123584+67269+126384+124032+68208+213452+64096+3n−65184+123248+12n−423364+12n−364276+6n−186844+6n−188496+64352+n−413924=211996n−130156.(3)
The neighborhood forgotten topological index ofHCPn is defined as follows:(21)FN∗G=∑st∈EGδs2δt2=∑st∈E1δs2δt2+∑st∈E2δs2δt2+∑st∈E3δs2δt2+⋯+∑st∈E21δs2δt2=122450+125849+125321+124361+217620+126272+127232+617092+1216132+128320+618180+226920+68192+3n−610368+126500+12n−426728+12n−368548+6n−1817288+6n−1819108+68720+n−427848=460640n−267604.(4)
The neighborhood second modifiedZagreb index of HCPn is defined as follows:(22)M2nmG=∑st∈EG1δsδt+∑st∈E21δsδt+∑st∈E31δsδt+⋯+∑st∈E211δsδt=12911225+1212380+1212240+1211960+217752+1213136+1213584+617296+1216384+1214032+618208+2113452+614096+3n−615184+1213248+12n−4213364+12n−3614176+6n−1816844+6n−1818496+614352+n−4113924=211996n−130156.(5)
The third and fifth NDe index ofHCPn is defined as follows:(23)ND3G=∑st∈EGδsδtδs+δt==∑st∈E1δsδtδs+δt+∑st∈E2δsδtδs+δt+∑st∈E3δsδtδs+δt+⋯+∑st∈E21δsδtδs+δt=12122570+122380103+12224099+12196091+27752182+123136112+123684120+67296178+126384677+124032128+68208186+213452232+64096128+3n−65184144+123248114+12n−423364116+12n−364176130+6n−186844176+6n−188496190+64352132+n−413924236=33635504n+8961688,ND5G=∑st∈EGδsδt+δtδs==∑st∈E1δsδt+δtδs+∑st∈E2δsδt+δtδs+∑st∈E3δsδt+δtδs+⋯+∑st∈E21δsδt+δtδs=122+122.457563025+1253212240+128940+244051938+122+1211356+642731824+1240331596+1213063+6505228+267303363+62+3n−62+121625812+12n−422+12n−3621371044+6n−1843221711+6n−1847772124+6545272+n−42=32.28529324+85.21361777n.(6)
The neighborhood Harmonic index ofHCPn is defined as follows:(24)NHG=∑st∈EG2δs+δt=∑st∈E12δs+δt+∑st∈E22δs+δt+∑st∈E32δs+δt+⋯+∑st∈E212δs+δt+12135+122103+12299+12291+2191+12156+12160+6189+12185+12164+6193+21116+6164+3n−6172+12157+12n−42158+12n−36165+6n−18188+6n−18195+6166+n−41118=0.5729928922n+0.6020913479.(7)
The neighborhood inverse sum index ofHCPn is defined as follows:(25)NIG=∑st∈EGδsδtδs+δt=∑st∈E1δsδtδs+δt+∑st∈E2δsδtδs+δt+∑st∈E3δsδtδs+δt+⋯+∑st∈E21δsδtδs+δt=12352+122380103+12224099+1228013+2387691+1228+1244815+6364889+12319285+12632+6136831+2336358+632+3n−636+12162457+12n−4229+12n−36208865+6n−18171144+6n−18424895+6108833+n−459=1402.089842n−347.4039608.(8)
The Sanskruti index ofHCPn is defined as follows:(26)NIG=∑st∈EGδsδtδs+δt−23=∑st∈E1δsδtδs+δt−23+∑st∈E2δsδtδs+δt−23+∑st∈E3δsδtδs+δt−23+⋯+∑st∈E21δsδtδs+δt−23=1235×3535+35−23+1235×6835+68−23+1235×6435+64−23+1235×5635+56−23+268×11468+114−23+1256×5656+56−23+1256×6456+64−23+664×11464+114−23+1256×11456+114−23+1256×7256+72−23+6114×72114+72−23+2114×118114+118−23+664×6464+64−23+3n−672×7272+72−23+1258×5658+56−23+12n−4258×5858+58−23+12n−3658×7258+72−23+6n−1858×11858+118−23+6n−18118×72118+72−23+668×6468+64−23n−4118×118118+118−23=2000631.573n−1847249.828.Table 4
Edge partition ofHCPn based on neighborhood degree of VHCPn
Edge representationNs,NtCardinality ofNs,NtE135,3512E235,6812E335,6412E435,5612E568,1142E656,5612E756,6412E864,1146E956,11412E1056,7212E11114,726E12114,1182E1364,646E1472,723n−6E1558,5612E1658,5812n−42E1758,7212n−36E1858,1186n−18E19118,726n−18E2068,646E21118,118n−4


## 3.1. Degree Based

We utilize Table2 to compute closed forms for the defined descriptors dependent on degree.(1)
First Zagreb index ofHCPn is defined as follows:(2)M1G=∑s∈VGds+dt=1210+1215+1213+3n16+618+12n−1814+12n−1215+6n−621+224+n−228+6n−622+1212=634n−10.(2)
Second Zagreb index ofHCPn is defined as follows:(3)M2G=∑st∈EGdsdt=1225+1250+1240+3n64+680+12n−1849+12n−1256+6n−698+2140+n−2196+6n−6112+1235=2908n−646.(3)
General Randić index ofHCPn is defined as follows:(4)RαG=∑st∈EGdsdtα=12dsdtα+12dsdtα+12dsdtα+3ndsdtα+6dsdtα+12n−18dsdtα+12n−12dsdtα+6n−6dsdtα+2dsdtα+n−2dsdtα+6n−6dsdtα+12dsdtα.(5)R1G=∑st∈EGdsdt1=12251+12501+12401+3n641+6801+12n−18491+12n−12561+6n−6981+21401+n−21961+6n−61121+12351=M2G.(6)R−1G=∑st∈EGdsdt−1=1225−1+1250−1+1240−1+3n64−1+680−1+12n−1849−1+12n−1256−1+6n−698−1+2140−1+n−2196−1+6n−6112−1+1235−1=0.7455102041+0.5790816327n.(7)R12G=∑st∈EGdsdt12=122512+125012+124012+3n6412+68012+12n−184912+12n−125612+6n−69812+214012+n−219612+6n−611212+123512=2.375607208+334.6947784n.(8)R−12G=∑st∈EGdsdt−12=1225−12+1250−12+1240−12+3n64−12+680−12+12n−1849−12+12n−1256−12+6n−698−12+2140−12+n−2196−12+6n−6112−12+1235−12=3.371752924+4.937319973n.(3a)
Whenα=1,(3b)
Whenα=−1,(3c)
Whenα=1/2,(3d)
Whenα=−1/2,(4)
Atom bond connectivity index ofHCPn is defined as follows:(9)ABCG=∑st∈EGds+dt−2dsdt=12825+121350+121140+3n732+615+12n−181249+12n−121356+6n−61998+21170+n−21398+6n−6528+1227=8.495057575+18.66489626n.(5)
The geometric arithmetic index ofHCPn is defined as follows:(10)GAG=∑st∈EG2dsdtds+dt=1222510+1225015+1224013+3n26416+628018+12n−1824914+12n−1225615+6n−629821+2214024+n−2219628+6n−6211222+1223512=39.4027062n+11.35413853.(6)
Exponential Reduced Zagreb index ofHCPn is defined as follows:(11)eRM2G=∑st∈EGeds−1dt−1=12e16+12e36+12e28+3ne49+6e63+12n−18e36+12n−12e42+6n−6e78+2e117+n−2e169+6n−6e91+12e24=2.487524928×1073n−4.975049857×1073.Table 2
Edge partition ofHCPn based on degree of VHCPn
Edge representationds,dtCardinalityofds,dtE15,512E25,1012E35,812E48,83nE58,106E67,712n−18E77,812n−12E87,146n−6E910,142E1014,14n−2E118,146n−6E125,712


## 3.2. Reverse Degree Based

We utilize Table3 to compute closed forms for the defined descriptors dependent on reverse degree of a vertex set of HCPn. where maximum degree for HCPn=ΔHCPn=14, which we use to compute reverse degree for the entire vertex set of lattice.(1)
Reverse Randić index ofHCPn is as follows where α=±1,±1/2:(1a)
Whenα=1,(12)ℝR1G=∑st∈EGℝsℝt1=∑st∈E1ℝsℝt+∑st∈E2ℝsℝt+⋯+∑st∈E12ℝsℝt=1210×10+1210×5+1210×7+3n7×7+67×5+12n−188×8+12n−128×7+6n−68×1+25×1+n−21×1+6n−67×1+1210×8=1904+1678n.(1b)
Whenα=−1,(13)ℝR−1G=∑st∈EGℝsℝt−1=∑st∈E1ℝsℝt−1+∑st∈E2ℝsℝt−1+⋯+∑st∈E12ℝsℝt−1=1210×10−1+1210×5−1+1210×7−1+3n7×7−1+67×5−1+12n−188×8−1+12n−128×7−1+6n−68×1−1+25×1−1+n−21×1−1+6n−67×1−1+1210×8−1=2407n784−159595600.(1c)
Whenα=1/2,(14)ℝR12G=∑st∈EGℝsℝt12=∑st∈E1ℝsℜt12+∑st∈E2ℝsℝt12+⋯+∑st∈E12ℝsℝt12=1210×1012+1210×512+1210×712+3n7×712+67×512+12n−188×812+12n−128×712+6n−68×112+25×112+n−21×112+6n−67×112+1210×812=240.6448479n+183.9070466.(1d)
Whenα=−1/2,(15)ℝR−12G=∑st∈EGℝsℝt−12=∑st∈E1ℝsℝt−12+∑st∈E2ℝsℝt−12+⋯+∑st∈E12ℝsℝt−12=1210×10−12+1210×5−12+1210×7−12+3n7×7−12+67×5−12+12n−188×8−12+12n−128×7−12+6n−68×1−12+25×1−12+n−21×1−12+6n−67×1−12+1210×8−12=8.921246062n−3.503847478.(2)
Reverse atom bond connectivity index ofHCPn is defined as follows:(16)ℝABCG=∑st∈E1ℝs+ℝt−2ℝsℝt=∑st∈E2ℝs+ℝt−2ℝsℝt+∑st∈E3ℝs+ℝt−2ℝsℝt+⋯+∑st∈E12ℝs+ℝt−2ℝsℝt=1210+10−210×10+1210+5−210×5+1210+7−210×7+3n7+7−27×7+67+5−27×5+12n−188+8−28×8+12n−128+7−28×7+6n−68+1−28×1+25+1−25×1+n−21+1−21×1+6n−67+1−27×2+1210+8−2108=24.04625241n+1.759584801.(3)
Reverse geometric arithmetic index ofHCPn is defined as follows:(17)ℝGAG=∑st∈EG2ℝsℝtℝs+ℝt=∑st∈E12ℝsℝtℝs+ℝt+∑st∈E22ℝsℝtℝs+ℝt+∑st∈E32ℝsℝtℝs+ℝt+⋯+∑st∈E122ℝsℝtℝs+ℝt=12210×1010+10+12210×510+5+12210×710+7+3n27×77+7+627×57+5+12n−1828×88+8+12n−1228×78+7+6n−628×18+1+225×15+1+n−221×11+1+6n−627×17+1+12210×810+8=35.71316677n+14.74470034.(4)
Reverse hyper Zagreb index ofHCPn is defined as follows:(18)ℝHMG=∑st∈EGℝs+ℝt2=∑st∈E1ℝs+ℝt2+∑st∈E2ℝs+ℝt2+∑st∈E3ℝs+ℝt2+⋯+∑st∈E12ℝs+ℝt2=1210+102+1210+52+1210+72+3n7+72+67+52+12n−188+82+12n−128+72+6n−68+12+25+12+n−21+12+6n−67+12+1210+82=7234n+7606.Table 3
Edge partition ofHCPn based on reverse degree of VHCPn
Edge representationℝs,ℝtCardinality ofℝs,ℝtE110,1012E210,512E310,712E47,73nE57,56E68,812n−18E78,712n−12E88,16n−6E95,12E101,1n−2E117,16n−6E1210,812


## 3.3. Neighborhood Degree Based

We utilize Table4 to evaluate closed formulas for the neighborhood degree-based descriptors representing topological properties.(1)
Neighborhood version of first Zagreb index ofHCPn is defined as follows:(19)M1∗G=∑st∈EGδs+δt=∑st∈E1δs+δt+∑st∈E2δs+δt+∑st∈E3δs+δt+⋯+∑st∈E21δs+δt=1270+12103+1299+1291+2182+12112+12120+6178+12170+12128+6186+2232+6128+3n−6144+12114+12n−42116+12n−36130+6n−18176+6n−18190+6132+n−4236=4792+5816n.(2)
The neighborhood secondZagreb index of HCPn is defined as follows:(20)M2∗G=∑st∈EGδsδt=∑st∈E1δsδt+∑st∈E2δsδt+⋯+∑st∈E21δsδt=121225+122380122240+12196027752+123136+123584+67269+126384+124032+68208+213452+64096+3n−65184+123248+12n−423364+12n−364276+6n−186844+6n−188496+64352+n−413924=211996n−130156.(3)
The neighborhood forgotten topological index ofHCPn is defined as follows:(21)FN∗G=∑st∈EGδs2δt2=∑st∈E1δs2δt2+∑st∈E2δs2δt2+∑st∈E3δs2δt2+⋯+∑st∈E21δs2δt2=122450+125849+125321+124361+217620+126272+127232+617092+1216132+128320+618180+226920+68192+3n−610368+126500+12n−426728+12n−368548+6n−1817288+6n−1819108+68720+n−427848=460640n−267604.(4)
The neighborhood second modifiedZagreb index of HCPn is defined as follows:(22)M2nmG=∑st∈EG1δsδt+∑st∈E21δsδt+∑st∈E31δsδt+⋯+∑st∈E211δsδt=12911225+1212380+1212240+1211960+217752+1213136+1213584+617296+1216384+1214032+618208+2113452+614096+3n−615184+1213248+12n−4213364+12n−3614176+6n−1816844+6n−1818496+614352+n−4113924=211996n−130156.(5)
The third and fifth NDe index ofHCPn is defined as follows:(23)ND3G=∑st∈EGδsδtδs+δt==∑st∈E1δsδtδs+δt+∑st∈E2δsδtδs+δt+∑st∈E3δsδtδs+δt+⋯+∑st∈E21δsδtδs+δt=12122570+122380103+12224099+12196091+27752182+123136112+123684120+67296178+126384677+124032128+68208186+213452232+64096128+3n−65184144+123248114+12n−423364116+12n−364176130+6n−186844176+6n−188496190+64352132+n−413924236=33635504n+8961688,ND5G=∑st∈EGδsδt+δtδs==∑st∈E1δsδt+δtδs+∑st∈E2δsδt+δtδs+∑st∈E3δsδt+δtδs+⋯+∑st∈E21δsδt+δtδs=122+122.457563025+1253212240+128940+244051938+122+1211356+642731824+1240331596+1213063+6505228+267303363+62+3n−62+121625812+12n−422+12n−3621371044+6n−1843221711+6n−1847772124+6545272+n−42=32.28529324+85.21361777n.(6)
The neighborhood Harmonic index ofHCPn is defined as follows:(24)NHG=∑st∈EG2δs+δt=∑st∈E12δs+δt+∑st∈E22δs+δt+∑st∈E32δs+δt+⋯+∑st∈E212δs+δt+12135+122103+12299+12291+2191+12156+12160+6189+12185+12164+6193+21116+6164+3n−6172+12157+12n−42158+12n−36165+6n−18188+6n−18195+6166+n−41118=0.5729928922n+0.6020913479.(7)
The neighborhood inverse sum index ofHCPn is defined as follows:(25)NIG=∑st∈EGδsδtδs+δt=∑st∈E1δsδtδs+δt+∑st∈E2δsδtδs+δt+∑st∈E3δsδtδs+δt+⋯+∑st∈E21δsδtδs+δt=12352+122380103+12224099+1228013+2387691+1228+1244815+6364889+12319285+12632+6136831+2336358+632+3n−636+12162457+12n−4229+12n−36208865+6n−18171144+6n−18424895+6108833+n−459=1402.089842n−347.4039608.(8)
The Sanskruti index ofHCPn is defined as follows:(26)NIG=∑st∈EGδsδtδs+δt−23=∑st∈E1δsδtδs+δt−23+∑st∈E2δsδtδs+δt−23+∑st∈E3δsδtδs+δt−23+⋯+∑st∈E21δsδtδs+δt−23=1235×3535+35−23+1235×6835+68−23+1235×6435+64−23+1235×5635+56−23+268×11468+114−23+1256×5656+56−23+1256×6456+64−23+664×11464+114−23+1256×11456+114−23+1256×7256+72−23+6114×72114+72−23+2114×118114+118−23+664×6464+64−23+3n−672×7272+72−23+1258×5658+56−23+12n−4258×5858+58−23+12n−3658×7258+72−23+6n−1858×11858+118−23+6n−18118×72118+72−23+668×6468+64−23n−4118×118118+118−23=2000631.573n−1847249.828.Table 4
Edge partition ofHCPn based on neighborhood degree of VHCPn
Edge representationNs,NtCardinality ofNs,NtE135,3512E235,6812E335,6412E435,5612E568,1142E656,5612E756,6412E864,1146E956,11412E1056,7212E11114,726E12114,1182E1364,646E1472,723n−6E1558,5612E1658,5812n−42E1758,7212n−36E1858,1186n−18E19118,726n−18E2068,646E21118,118n−4


## 4. Conclusion

From Table5, we can exactly say that all the descriptors behave positively except exponential reduced Zagreb index and shown in Table 5 by symbol +and− as the value n increases and the descriptor ND3 dominates amongst all the topological descriptors which assure us about most dominant topological property for HCPn, hexagonal closed-packed crystal structure lattice. We compute all the above calculated descriptors for n=1, i.e., for a unit hexagonal close-packed crystal cell.Table 5
Analyzing descriptors.
DescriptorsGeneral closed formula for hexagonal crystal structure latticen=1=one unit cellCheck for positivity+,negativity−,and dominationM1HCPn643n−10633(+)R1HCPn=M2HCPn2908n−6462262(+)R−1HCPn0.579n+0.74551.3245(+)R1/2HCPn334.69n+2.38337.07(+)R−1/2HCPn4.94n+3.378.31(+)ABCHCPn18.66n+8.49527.155(+)GAHCPn39.4027n+11.35450.7567(+)eRM2G2.4875×1073n−4.975×1073-2.487524929×1073(−)ℝR1HCPn1678n+19043582(+)ℝR−1HCPn2407/784n−15959/56000.2203316(+)ℝR1/2HCPn240.6448n+183.907424.5518(+)ℝR−1/2HCPn8.92n−3.505.42(+)ℝABCHCPn24.04n+1.759525.7995(+)ℝGAHCPn35.71n+14.744750.4547(+)ℝHMHCPn7234n+760614840(+)M1∗HCPn4792+5816n10608(+)M2∗HCPn211996n−13015681840(+)MN∗HCPn460640n−267604193036(+)M2nmHCPn211996n−13015681840(+)ND3HCPn33635504n+8961688 4259719242597192(+) and dominatesND5HCPn32.28529324+85.21361777n117.4989(+)NHHCPn0.5729928922n+0.60209134791.17508424(+)NIHCPn1402.089842n−347.40396081054.685881(+)SHCPn2000631.573n−1847249.828153381.745(+)



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*Source: 6069666-2022-07-31.xml*