# Vertex-Edge-Degree-Based Topological Properties for Hex-Derived Networks

Authors: Ali Ahmad; Muhammad Imran
Journal: Complexity
 (2022)

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## Abstract

A topological index can be focused on uprising of a chemical structure into a real number. The degree-based topological indices have an active place among all topological indices. These topological descriptors intentionally associate certain physicochemical assets of the corresponding chemical compounds. Graph theory plays a very useful role in such type of research directions. The hex-derived networks have vast applications in computer science, physical sciences, and medical science, and these networks are constructed by hexagonal mesh networks. In this paper, we determined the exact values of vertex-edge-degree-based topological descriptors for hex-derived networksHDN1p and HDN2p, which are generated by the hexagonal network of dimension p.


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## Body

## 1. Introduction

The applications of topological descriptors of chemical structure are nowadays a normal process in the education of structure-property relations, specifically in QSPR/QSAR studies. Topological indices play a dynamic part in the QSPR/QSAR study. They associate certain physicochemical assets of chemical compounds. Graph theory has provided pharmacists with an assortment of suitable apparatuses, such as topological indices. Chemicals and chemical compounds are frequently displayed by chemical graphs. A chemical graph is an illustration of the structural formula of a chemical compound in terms of graph theory, in which atoms are denoted with vertices and edges show the chemical bonding between them. Lately, a latest topic that has piqued the interest of researchers is cheminformatics, which is a composite of chemistry, information science, and mathematics, in which the QSAR/QSPR relationship, bioactivity, and classification of chemical compounds are investigated.The topological descriptor is a real number associated with chemical compositions that maintains the correlation of chemical structures with a variety of physicochemical properties, chemical reactivity, or biological activity. Topological indices are classified into three types: distance-based topological indices, degree-based topological indices, and counting-related topological indices. Numerous researchers have recently discovered topological indices for the study of fundamental properties of molecular graphs or networks. These networks have very appealing topological properties, which have been considered in various characteristics such as [1–8].Chen et al. [9] explained the construction of hexagonal mesh networks that consist of triangles, as shown in Figure 1. Furthermore, we gather the pth hexagonal mesh by putting p triangles around the boundary of each hexagon. Imran et al. defined the new hex-derived networks, namely, first type HDN1p (see Figure 2) and second type HDN2p (see Figure 3); for detailed construction, see [10]. Simonraj and George [11] created the new network which is named as third type of hex-derived networks. Koam et al. [12] computed the vertex-edge-based topological indices of some hex-derived networks. There are some works related to hex-derived networks which can be seen in [13–15]. Related research papers that contain the theoretical as well as application aspects for new research directions can be found in [16–22].Figure 1
Hexagonal meshes: (a)HX2 and (b)HX3.
(a)(b)Figure 2
Hex-derived networkHDN1p for p=3.Figure 3
Hex-derived networkHDN2p for p=4.


## 2. Preliminaries

LetG=V,E be a simple connected graph with E being the edge set and V being the vertex set. The symbol Ψθ denoted the concept of degree of a vertex θ, and it is defined by the number of attached edges with θ. The symbol Nθ denoted the number of all vertices adjacent to θ and is called as the open neighborhood of a vertex θ. On the contrary, the symbol Nθ is the union of θ and Nθ and called as the closed neighborhood of θ. The concept of ve-degree denoted by Ψveθ, and can be defined as follows: for any vertex θ from the vertex set of a graph, is the number of different edges that are attached to any vertex from the closed neighborhood of θ. In this research work, we elaborated different ve-degree-associated topological descriptors. In [23], vertex/edge-degree-based topological indices are defined in which they computed the degree of an edge uv as du+dv−2. In this article, we consider ve-degree which is the degree of a vertex and is calculated by adding the degrees of its all-neighboring vertices.(1)ΓG=∑θϑ∈EGϕΨveθ,Ψveϑ,ϕΨveθ,Ψveϑ=∑θϑ∈EΨveθ+Ψveϑ,firstve−degree ZagrebβindexMβve1,ϕΨveθ,Ψveϑ=∑θϑ∈EΨveθ+Ψveϑ,secondve−degree Zagreb indexMβve2,ϕΨveθ,Ψveϑ=∑θϑ∈E2Ψveθ+Ψveϑ,ve−degree harmonic indexHve,ϕΨveθ,Ψveϑ=∑θϑ∈EΨveθ×Ψveϑ−1/2,ve−degree Randic indexRve,ϕΨveθ,Ψveϑ=∑θϑ∈EΨveθ+Ψveϑ−1/2,ve−degree sum−connectivity indexχve,ϕΨveθ,Ψveϑ=∑θϑ∈EΨveθ+Ψveϑ−2Ψveθ×Ψveϑ1/2,ve−degree atom−bond connectivity indexABCve,ϕΨveθ,Ψveϑ=∑θϑ∈E2Ψveθ×Ψveϑ1/2Ψveθ+Ψveϑ,ve−degree geometric−arithmetic indexGAve.The researchers in [24–32] detailed different ve-degree topological invariants. This research work contains the computational exact results of given above descriptors.


## 3. Hex-Derived NetworkHDN1p

LetHDN1p be the notation for the hex-derived network of the first type, and it is shown in Figure 2. The original hex-derived network HDN1p contains 9p2−15p+7 total number of vertices in which there are 6p2−12p+6 vertices of degree 3, 6 vertices of degree 5, 6p−12 vertices of degree 7, and 3p2−9p+7 vertices of degree 12. There are 27p2−51p+24 count of edges for the graph HDN1p; all these edges are partitioned into eight subsets according to their degrees and corresponding ve-degrees of end vertices elaborated in equations (9)–(16).LetΞveγ,δα,β be the edge partition of HDN1p according to its degrees α,β and ve-degrees γ,δ. It is defined as(2)Ξveγ,δα,β=θϑ∈EHDN1p:dθ=α,dϑ=β,Ψθ=γ,Ψϑ=δ.AndΞveγ,δα,β represents the number of edges in Ξveγ,δα,β.(3)Ξveγ,δ3,5=12,forγ=24,δ=32,(4)Ξveγ,δ3,7=12ifγ=24,δ=45,12ifγ=31,δ=45,12ifγ=26,δ=45,12p−48ifγ=26,δ=47,6p−24ifγ=31,δ=47,(5)Ξveγ,δ3,12=12ifγ=24,δ=73,12ifγ=31,δ=73,6p−18ifγ=26,δ=80,12p−36ifγ=31,δ=80,12ifγ=36,δ=73,18p−54ifγ=36,δ=80,18p2−90p+114ifγ=36,δ=90,(6)Ξveγ,δ5,7=12ifγ=32,δ=45,(7)Ξveγ,δ5,12=6ifγ=32,δ=73,(8)Ξveγ,δ7,7=12ifγ=45,δ=47,6p−30ifγ=47,δ=47,(9)Ξveγ,δ7,12=12ifγ=45,δ=73,12ifγ=45,δ=80,12p−48ifγ=47,δ=80,(10)Ξveγ,δ4,18=12ifγ=73,δ=80,6ifγ=73,δ=90,6p−24ifγ=80,δ=80,12p−36ifγ=80,δ=90,9p2−51p+72ifγ=90,δ=90.Given below are someve-degree-based indices, for example, Mβve1 index, Mve2 index, Rve index, ABCve index, GAve index, Hve index, and χve index for HDN1p.Theorem 1.
LetHDN1p be the first type of hex-derived network; then,(i)
Mβve1HDN1p=3888p2−10032p+6276(ii)
Mve2HDN1p=131220p2−404040p+298614(iii)
HveHDN1p=27p2/70−312850210871353p/920029146474984+1630411987161294780214346433/19563734063325532742904992750Proof.
LetHDN1p be the notation for the hex-derived network of the first type, and it is shown in Figure 2. The original hex-derived network HDN1p contains 9p2−15p+7 total number of vertices in which there are 6p2−12p+6 vertices of degree 3, 6 vertices of degree 5, 6p−12 vertices of degree 7, and 3p2−9p+7 vertices of degree 12. There are 27p2−51p+24 count of edges for the graph HDN1p; all these edges are partitioned into eight subsets according to their degrees and relative ve-degrees of both end vertices elaborated in equations (9)–(16). Evaluating equation (2), we can determine the first ve-degree Zagreb β index as(11)Mβve1HDN1p=Ξveγ,δ3,5γ+δ+Ξveγ,δ3,7γ+δ+Ξveγ,δ3,12γ+δ+|Ξveγ,δ5,7|γ+δ+|Ξveγ,δ5,12|γ+δ+Ξveγ,δ7,7γ+δ+Ξveγ,δ7,12γ+δ+Ξveγ,δ12,12γ+δ.
Evaluating equations (9)–(16) and after simplifications, we will have the required results as(12)Mβve1HDN1p=3888p2−10032p+6276.
Evaluating equation (3), we can compute the second ve-degree Zagreb index as(13)Mve2HDN1p=Ξveγ,δ3,5γ×δ+Ξveγ,δ3,7γ×δ+Ξveγ,δ3,12γ×δ+Ξveγ,δ5,7γ×δ+Ξveγ,δ5,12γ×δ+Ξveγ,δ7,7γ×δ+Ξveγ,δ7,12γ×δ+Ξveγ,δ12,12γ×δ.
Evaluating equations (9)–(16) and after simplifications, we will have the required result as(14)Mve2HDN1p=131220p2−404040p+298614.
Evaluating equation (4), we can determine the ve-degree harmonic index as(15)HveHDN1p=Ξveγ,δ3,52γ+δ+Ξveγ,δ3,72γ+δ+Ξveγ,δ3,122γ+δ+Ξveγ,δ5,72γ+δ+Ξveγ,δ5,122γ+δ+Ξveγ,δ7,72γ+δ+Ξveγ,δ7,122γ+δ+Ξveγ,δ12,122γ+δ.
Evaluating equations (9)–(16) and after simplifications, we get(16)HveHDN1p=27p270−312850210871353p920029146474984+163041198716129478021434643319563734063325532742904992750.Theorem 2.
LetHDN1p be the first type of hex-derived network; then,(i)
RveHDN1p=10/10+1/10p2+−2053/5640+61222/611+2/10+61457/1457+35/20+3155/155+3235/235−10/2+3130/260p+122263/2263−241222/611−32/10−241457/1457−95/20−155/31−8235/235+1110/15−130/260+3/4+438/73+29/470+273/73+730/365+3146/292+7365/365+30/15(ii)
χveHDN1p=35/10+314/7p2+310/20+1273/73+6170/85+78/13−175/10+929/29+394/47+4111/37+12127/127−1514/7+3106/53p−310/5+2214/7+619/19+326/13−2729/29+623/23+417/17+725/25−48127/127−9106/53+6118/59+1277/77+1271/71+469/23+1297/97−18170/85−478/13−1594/47+12109/109+6163/163+2105/35−12111/37−4873/73Proof.
Theve-degree Randic index can be determined by evaluating the edge partitions in equation (5):(17)RveHDN1p=Ξveγ,δ3,5γ×δ−12+Ξveγ,δ3,7γ×δ−1/2+Ξveγ,δ3,12γ×δ−1/2+Ξveγ,δ5,7γ×δ−1/2+Ξveγ,δ5,12γ×δ−1/2+Ξveγ,δ7,7γ×δ−1/2+Ξveγ,δ7,12γ×δ−1/2+Ξveγ,δ12,12γ×δ−1/2.
The methodology of edge partitions can be determined from equations (9)–(16); after mathematical calculations, we get the following result:(18)RveHDN1p=1010+110p2+−20535640+61222611+210+614571457+3520+3155155+3235235−102+3130260p+1222632263−241222611−3210−2414571457−9520−15531−8235235+111015−130260+34+43873+29470+27373+730365+3146292+7365365+3015.
Theve-degree sum-connectivity index can be determined by using the values from equations (9)–(16) in equation (6); we get the following:(19)χveHDN1p=Ξveγ,δ3,5γ+δ−1/2+Ξveγ,δ3,7γ+δ−1/2+Ξveγ,δ3,12γ+δ−1/2+Ξveγ,δ5,7γ+δ−1/2+Ξveγ,δ5,12γ+δ−1/2+Ξveγ,δ7,7γ+δ−1/2+Ξveγ,δ7,12γ+δ−1/2+Ξveγ,δ12,12γ+δ−1/2.
After simplification, we obtain(20)χveHDN1p=3510+3147p2+31020+127373+617085+7813−17510+92929+39447+411137+12127127−15147+310653p−3105+22147+61919+32613−272929+62323+41717+72525−48127127−910653+611859+127777+127171+46923+129797−1817085−47813−159447+12109109+6163163+210535−1211137−487373.Theorem 3.
LetHDN1p be the third type of hex-derived network; then,(i)
ABCveHDN1p=178/10+310/5p2+−17178/30+3570/20+1223/47+316895/155+1547/47−310+35/5+3158/40+686762/611+221/5+1227683/1457p−2486762/611+28970/65+12230826/2263−95/5+92/4+1294/47+41610/73+30/2−6023/47+19310/15+810585/365+411470/155+2010/15−3158/10−621/5−4827683/1457+4178/5+123/5−9570/20−6047/47+27811/73+117530/365+315038/292+355115/365−916895/155Proof.
The numerical descriptor of theve-degree atom-bond connectivity index can be calculated by evaluating the values of edge partitions in equation (7):(21)ABCveHDN1p=Ξveγ,δ3,5γ+δ−2γ×δ1/2+Ξveγ,δ3,7γ+δ−2γ×δ1/2+Ξveγ,δ3,12γ+δ−2γ×δ1/2+Ξveγ,δ5,7γ+δ−2γ×δ1/2+Ξveγ,δ5,12γ+δ−2γ×δ1/2+Ξveγ,δ7,7γ+δ−2γ×δ1/2+Ξveγ,δ7,12γ+δ−2γ×δ1/2+Ξveγ,δ12,12γ+δ−2γ×δ1/2.
Evaluating equations (9)–(16) and after mathematical calculations, we will have(22)ABCveHDN1p=17810+3105p2+−1717830+357020+122347+316895155+154747−310+355+315840+686762611+2215+12276831457p−2486762611+2897065+122308262263−955+924+129447+4161073+302−602347+1931015+810585365+411470155+201015−315810−6215−48276831457+41785+1235−957020−604747+2781173+117530365+315038292+355115365−916895155.
Theve-degree geometric-arithmetic index can be calculated by evaluating the methodology of edge partitions in (1):(23)GAveHDN1p1=Ξveγ,δ3,52γ×δ1/2γ+δ+Ξveγ,δ3,72γ×δ1/2γ+δ+Ξveγ,δ3,122γ×δ1/2γ+δ+Ξveγ,δ5,72γ×δ1/2γ+δ+Ξveγ,δ5,122γ×δ1/2γ+δ+Ξveγ,δ7,72γ×δ1/2γ+δ+Ξveγ,δ3,122γ×δ1/2γ+δ+Ξveγ,δ12,122γ×δ1/2γ+δ+Ξveγ,δ5,122γ×δ1/2γ+δ+Ξveγ,δ7,72γ×δ1/2γ+δ+Ξveγ,δ7,122γ×δ1/2γ+δ+Ξveγ,δ12,122γ×δ1/2γ+δ.
Evaluating equations (9)–(16) and after mathematical calculations, we will have(24)GAveHDN1p=36107+9p2+−39+24122273+144217+2145713+216529+3215537+96235127−180107+2413053p+3226313−96122273−432217−8145713−648529−1158155703−65462352921+27961077−12961303763+4837+4843897+14473109+36730163+1614635+37243653009+483023+73825.


## 4. Hex-Derived NetworkHDN2p

LetHDN2p be the notation of the hex-derived network of the second type, and it is shown in Figure 3. This hex-derived network HDN2p contained 9p2−15p+7 total number of vertices in which there are 6p vertices of degree 5, 6p2−18p+12 vertices of degree 6, 6p−12 vertices of degree 7, and 3p2−9p+7 vertices of degree 12.The total number of edges of this network is36p2−72p+36, and they can be partitioned into ten different subsets according to their degrees and associated with ve-degrees of both end vertices, which are shown in equations (17)–(26). Given below are some ve-degree-based indices, for example, Mβve1 index, Mve2 index, Rve index, ABCve index, GAve index, Hve index, and χve index for the HDN2p network. Let Ξveγ,δα,β be the edge partition of HDN2p according to its degrees α,β and ve-degrees γ,δ. It is defined as(25)Ξveγ,δα,β=θϑ∈EHDN2p:dθ=α,dϑ=β,Ψθ=γ,Ψϑ=δ.AndΞveγ,δα,β represents the number of edges in Ξveγ,δα,β.(26)Ξveγ,δ12,12=12ifγ=89,δ=97,6ifγ=89,δ=108,6p−34ifγ=97,δ=97,12p−36ifγ=97,δ=108,9p2−51p+72ifγ=108,δ=108,(27)Ξveγ,δ7,12=12ifγ=52,δ=89,12ifγ=52,δ=97,12p−48ifγ=54,δ=97,(28)Ξveγ,δ7,7=12ifγ=52,δ=54,12p−30ifγ=54,δ=54,(29)Ξveγ,δ6,12=12ifγ=47,δ=89,12p−36ifγ=47,δ=97,12ifγ=54,δ=89,18p−54ifγ=54,δ=97,18p2−90p+114ifγ=54,δ=108,(30)Ξveγ,δ6,7=12ifγ=47,δ=52,6p−24ifγ=47,δ=54,(31)|Ξveγ,δ6,6=6p−12ifγ=47,δ=54,9p2−39p+42ifγ=54,δ=54,(32)Ξveγ,δ5,12=6ifγ=36,δ=89,12ifγ=35,δ=89,6p−18ifγ=38,δ=97,(33)Ξveγ,δ5,7=12ifγ=36,δ=52,12ifγ=35,δ=52,12p−48ifγ=38,δ=54,12ifγ=38,δ=52,(34)Ξveγ,δ5,6=12ifγ=35,δ=47,12p−36ifγ=38,δ=47,(35)Ξveγ,δ5,5=6ifγ=35,δ=35,12ifγ=35,δ=36.Theorem 4.
LetHDN2p be the second type of the hex-derived network; then,(i)
Mβve1HDN2p=5832p2−15132p+9528(ii)
Mve2HDN2p=236196p2−723330p+532842(iii)
HveHDN2p=17p2/36−2483413922281p/4268774116260+162959160788410914818964603487/747847045658086922387267678250Proof.
LetHDN2p be the second type of the hex-derived network which is shown in Figure 3. The hex-derived network HDN2p has 9p2−15p+7 vertices. There are 36p2−72p+36 number of edges of HDN2p which are partitioned into ten partitions based on their degrees and corresponding ve-degrees of end vertices given in equations (17)–(26).
The firstve-degree Zagreb β index can be calculated as(36)Mβve1HDN2p=Ξveγ,δ12,12γ+δ+Ξveγ,δ7,12γ+δ+Ξveγ,δ7,7γ+δ+Ξveγ,δ6,12γ+δ+Ξveγ,δ6,7γ+δ+Ξveγ,δ6,6γ+δ+Ξveγ,δ5,12γ+δ+|Ξveγ,δ5,7|γ+δ+Ξveγ,δ5,6γ+δ+Ξveγ,δ5,5γ+δ.
Evaluating equations (17)–(22) and after simplifications, we will have the final result as follows:(37)Mβve1HDN2p=5832p2−15132p+9528.
The secondve-degree Zagreb index can be calculated as(38)Mve2HDN2p=Ξveγ,δ12,12γ×δ+Ξveγ,δ7,12γ×δ+|Ξveγ,δ7,7|γ×δ+Ξveγ,δ6,12γ×δ+Ξveγ,δ6,7γ×δ+Ξveγ,δ6,6γ×δ+Ξveγ,δ5,12γ×δ+Ξveγ,δ5,7γ×δ+Ξveγ,δ5,6γ×δ+Ξveγ,δ5,5γ×δ.
Evaluating equations (17)–(22) and after simplifications, we will have the final result as follows:(39)Mve2HDN2p=236196p2−723330p+532842.
Theve-degree harmonic index can be determined as(40)HveHDN2p=Ξveγ,δ12,122γ+δ+Ξveγ,δ7,122γ+δ+Ξveγ,δ7,72γ+δ+Ξveγ,δ6,122γ+δ+Ξveγ,δ6,72γ+δ+Ξveγ,δ6,62γ+δ+Ξveγ,δ5,122γ+δ+Ξveγ,δ5,72γ+δ+Ξveγ,δ5,62γ+δ+Ξveγ,δ5,52γ+δ.
Evaluating equations (17)–(22) and after some calculations, we have the final result as follows:(41)HveHDN2p=17p236−2483413922281p4268774116260+162959160788410914818964603487747847045658086922387267678250.Theorem 5.
LetHDN2p be the second type of the hex-derived network; then,(i)
RveHDN2p=1/4+2/6p2+124559/4559+2282/141−52/6+33686/1843+2291/291+5582/291+257/57+61786/893−1189/1164p+2534/267−2282/47+124183/4183+128633/8633+123115/3115−857/57+3494/247−93686/1843−364559/4559+24838/30555+6611/611+267/267+61157/1157+6455/455+13/13+121645/1645−181786/893+235/35+192/18+89/89−2291/97+61261/1261+78/39−17582/291(ii)
χveHDN2p=3/2+2+6/4p2+1+30151/151−113/6−52+12101/101−176/12+623/23+1285/85+215/15+12205/205+3194/97p+411/11+334/17+26−2423/23−3+631/31+3/1122+210/5−215/5−36205/205+12149/149+6106/53+1271/71−12194/97−102151/151+12143/143−36101/101+6197/197+4141/47+2186/31+487/29+682/41−3685/85+370/35+23/3+65/25+192/3Proof.
Theve-degree Randic index is measured by evaluating equations (17)–(26) in the following:(42)RveHDN2p=Ξveγ,δ12,12γ×δ−1/2+Ξveγ,δ7,12γ×δ−1/2+Ξveγ,δ7,7γ×δ−1/2+Ξveγ,δ6,12γ×δ−1/2+Ξveγ,δ6,7γ×δ−1/2+Ξveγ,δ6,6γ×δ−1/2+Ξveγ,δ5,12γ×δ−1/2+Ξveγ,δ5,7γ×δ−1/2+Ξveγ,δ5,6γ×δ−1/2+Ξveγ,δ5,5γ×δ−1/2.
After simplification, we obtain(43)RveHDN2p=14+26p2+1245594559+2282141−526+336861843+2291291+5582291+25757+61786893−11891164p+2534267−228247+1241834183+1286338633+1231153115−85757+3494247−936861843−3645594559+2483830555+6611611+267267+611571157+6455455+1313+1216451645−181786893+23535+19218+8989−229197+612611261+7839−17582291.
Theve-degree sum-connectivity index is measured by evaluating equations (17)–(26) in the given following formula:(44)χveHDN2p=Ξveγ,δ12,12γ+δ−1/2+Ξveγ,δ7,12γ+δ−1/2+Ξveγ,δ7,7γ+δ−1/2+Ξveγ,δ6,12γ+δ−1/2+Ξveγ,δ6,7γ+δ−1/2+Ξveγ,δ6,6γ+δ−1/2+Ξveγ,δ5,12γ+δ−1/2+Ξveγ,δ5,7γ+δ−1/2+Ξveγ,δ5,6γ+δ−1/2+Ξveγ,δ5,5γ+δ−12.
After simplification, we obtain(45)χveHDN2p=32+2+64p2+1+30151151−1136−52+12101101−17612+62323+128585+21515+12205205+319497p+41111+33417+26−242323−3+63131+31122+2105−2155−36205205+12149149+610653+127171−1219497−102151151+12143143−36101101+6197197+414147+218631+48729+68241−368585+37035+233+6525+1923.Theorem 6.
LetHDN2p be the second type of the hex-derived network; then, (i)
ABCveHDN2p=214/12+45/3+106/6p2+23102/47−17214/36−11106/18+2570/19−205/3+6148238/893+31358/97+12647378/4559+259073/291+483/97+586718/291p−63102/47+12560522/4183−8570/19+12380030/3115+122717/247−18148238/893−91358/97−36647378/4559+1217/35+2106/9−1883/291+765/9−259073/97+423783/1261+22415/35−1786718/291+1118/13+10947/89+6160823/1157+24397118/8633+61547/91+48329/329+28366/89+659267/611+2214/3+5785/89(ii)
GAveHDN2p=18+122p2+−78+72282/101−602+43686/45+4559/6+144291/205+180582/151+3657/23+241786/85p+762+7289/125−432291/205+481261/149+7278/53−612582/151+66+72534/143−216282/101+34183/17+48633/31+63115/31−14457/23+8494/15−43686/15−4559/2+16611/33+72267/197+161157/47+16455/29+3613/11+121645/41−721786/85+14435/71Proof.
Theve-degree atom-bond connectivity index is measured as(46)ABCveHDN2p=Ξveγ,δ12,12γ+δ−2γ×δ1/2+Ξveγ,δ7,12γ+δ−2γ×δ1/2+Ξveγ,δ7,7γ+δ−2γ×δ1/2+Ξveγ,δ6,12γ+δ−2γ×δ1/2+Ξveγ,δ6,7γ+δ−2γ×δ1/2+Ξveγ,δ6,6γ+δ−2γ×δ1/2+Ξveγ,δ5,12γ+δ−2γ×δ1/2+Ξveγ,δ5,7γ+δ−2γ×δ1/2+Ξveγ,δ5,6γ+δ−2γ×δ1/2+Ξveγ,δ5,5γ+δ−2γ×δ1/2.
By putting the values from equations (17)–(26) and after simplification, we obtain(47)ABCveHDN2p=21412+453+1066p2+2310247−1721436−1110618+257019−2053+6148238893+3135897+126473784559+259073291+48397+586718291p−6310247+125605224183−857019+123800303115+122717247−18148238893−9135897−366473784559+121735+21069−1883291+7659−25907397+4237831261+2241535−1786718291+111813+1094789+61608231157+243971188633+6154791+48329329+2836689+659267611+22143+578589.
Theve-degree geometric-arithmetic index is measured as(48)GAveHDN2p=Ξveγ,δ12,122γ×δ1/2γ+δ+Ξveγ,δ7,122γ×δ1/2γ+δ+Ξveγ,δ7,72γ×δ1/2γ+δ+Ξveγ,δ6,122γ×δ1/2γ+δ+Ξveγ,δ6,72γ×δ1/2γ+δ+Ξveγ,δ6,62γ×δ1/2γ+δ+Ξveγ,δ5,122γ×δ1/2γ+δ+Ξveγ,δ5,72γ×δ1/2γ+δ+Ξveγ,δ5,62γ×δ1/2γ+δ+Ξveγ,δ5,52γ×δ1/2γ+δ.
By putting the values from equations (17)–(22) and after simplification, we obtain(49)GAveHDN2p=18+122p2+−78+72282101−602+4368645+45596+144291205+180582151+365723+24178685p+762+7289125−432291205+481261149+727853−612582151+66+72534143−216282101+3418317+4863331+6311531−1445723+849415−4368615−45592+1661133+72267197+16115747+1645529+361311+12164541−72178685+1443571.


## 5. Conclusion

In this paper, certainve-degree-based topological indices, namely, first ve-degree Zagreb βMβve1 index, second ve-degree Zagreb Mve2 index, ve-degree harmonic Hve index, ve-degree Randic Rve index, ve-degree sum-connectivity χve index, ve-degree atom-bond connectivity ABCve index, and ve-degree geometric-arithmetic GAve index, for the first type of hex-derived network HDN1p and second type of hex-derived network HDN2p are studied. We calculated exact formulas of the aforementioned ve-degree-based topological indices for these derived networks. These results provide a basis to comprehend the deep network topology of these vital networks. The numerical comparison of the first type of hex-derived network HDN1p and second type of hex-derived network HDN2p is shown in Tables 1 and 2.Table 1
Numerical comparison ofMβve1, Mve2, Hve, Rve, χve, ABCve, and GAve for the first type of hex-derived network HDN1p.
pMβve1Mve2HveRveχveABCveGAve55331615589148.02608.524141.75287.893418.85686052259829411.92912.70363.506134.04649.967126564390011416.60317.71589.809189.87931.598174852546437422.04923.559120.67255.441263.89230916729107428.26630.233156.08330.691646.410294756938021435.25437.743196.02415.682079.7113663721173179443.01446.083240.52510.372563.2124457641434581451.54655.259289.56614.763097.5135329321722227460.84865.263343.16728.873682.5146278762036117470.92376.103401.30852.694317.7Table 2
Numerical comparison ofMβve1, Mve2, Hve, Rve, χve, ABCve, and GAve for the second type of hex-derived network HDN2p.
pMβve1Mve2HveRveχveABCveGAve57966828210929.11469.355550.648103.29559.926128688469591813.72714.09377.722158.57874.857189372704313619.28419.803110.59225.701259.78261720986274625.78626.484149.24304.641714.893457321315474833.23234.134193.67395.462239.3104414081691914241.62242.760243.88498.072834.1115487482115592850.95752.356299.89612.513498.8126677522586510661.23762.921361.68738.814233.2137984203104667672.46174.458429.25876.915037.7149407523670063884.62986.968502.601027.05912.3



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*Source: 7025480-2022-02-11.xml*