# Comparative Analysis of Subdivided Hex-Derived Networks Using Structural Descriptors

Authors: Maqsood Ahmad; Muhammad Javaid; Ebenezer Bonyah; Muhammad Saeed
Journal: Journal of Chemistry
 (2022)

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## Abstract

A structural descriptor is a numeric quantity, invariant under symmetry, extracted from a molecular graphΓ using tools from mathematics. Hex-derived networks HDN1n and HDN2n being dual of (3,12,12) and (4,8,8) carbon allotrope (nanocones, nanotubes) are quite fascinating structures in mathematical chemistry. In this paper, we compute exact formulas of various significant structure descriptors for the subdivision of the second and the third kind of hex-derived networks SHDN2n and SHDN3n, respectively. Moreover, we perform a comparative graphical analysis among certain descriptors of SHDN1n, SHDN2n, and SHDN3n to help understand and provide an insight to anticipate some physicochemical properties of the networks under discussion.


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## Body

## 1. Introduction

Graph theory achieved a remarkable place in mathematical chemistry, and the interdisciplinary branch thus realized as chemical graph theory (CGT). It provides tools to model molecules of certain chemical compounds mathematically to compute various structural descriptors to comprehend and envisage their structural and chemical properties [1, 2]. Recently, researchers have a growing interest and paid considerable attention to another rapidly developing interdisciplinary area called chemometrics. In chemometric analysis, the tools from mathematics, statistics, and information science are used for extracting maximum information from chemical data. Like, while performing QSPR/QSAR modeling, the multivariate regression analysis largely relies upon structural descriptors data. Consequently, we can extract useful information to predict the physical, biological, and chemical attributes of chemical species that give rise to the controlled design of new drugs and chemical or biological compounds.A topological (molecular) descriptor is a function denoted byTopΓ having domain ∑={all connected graphs} and range being the R (set of reals) along with following equality TopΓ1=TopΓ2 provided that Γ1 and Γ2 are isomorphic graphs. Numerous types of degree-based, distance-based, spectral-based, and polynomial-related descriptors of chemical graphs were established (IUPAC (International Union for Pure and Applied Chemistry)) to ascertain their applications in CGT. Among these classes, the degree-based descriptors are considered to be the most important and applicable in mathematical chemistry and there in biochemistry [3]. For example, among the most well-explored and well-known descriptors are the Randic′ indices [4], Wiener and Szeged indices [5, 6], Zagreb indices [7, 8], ABC [9], its fourth version ABC4 [10], GA [11], its fifth version GA5 [12], and the Hosoya Z indices [13]. The idea of using polynomials as an index was first conceived and implemented by Po′lya [14]. Later on, several graph polynomials were introduced in the literature that proved to be very imperative and played a key role in CGT such as matching polynomial [15], Zhang-Zhang polynomial [16], Schultz polynomial [17], and Tutt polynomial [18, 19]. M-polynomial provides us an alternate, efficient, and general way to compute exact formulas for various descriptors [20]. Recently, many researchers investigated and utilized various structural descriptors to the study of diverse properties of different molecular graphs such as 2-dimensional silicon-carbon [21], dendrimers [22], nanotubes [23], hydrocarbons [24], operational chemical structures [25], and synthetic polymers [26]. For further study on different chemical structures, we refer to [27–29].In [30], Chen et al. introduced the hexagonal network comprising of triangles, and such types of graphs are commonly known as oxide networks in mathematical chemistry. Figure 1 illustrates the self-explanatory construction of the hexagonal network of dimension 2 HX2, dimension 3 HX3, and dimension 4 HX4. Continuing in this way, by placing an outermost layer of n triangles, we can get the hexagonal network HXn with dimension n. From Figure 1, it is readily be observed that the hexagonal network having dimension one does not exist. Nocetti et al. investigated the mobile user tracking in hexagonal-based cellular networks [31].Figure 1
Hexagonal networks: (a) HX(2), (b) HX(3), and (c) HX(4).
(a)(b)(c)Hex-derived networks are, basically, mathematical objects and originate fromtessellation (geometry of tiles). Second of all, these networks (stiffest elastic networks) are also a topic of study in civil engineering, when edges are considered to be beams, under the name Triakis tilingHDN1 and kisrhombille tilingHDN2. On the other hand, these objects are subject of study in symmetry groups. Moreover, these networks are used in computer sciences as mesh networking of computers for cost-effectiveness or data efficacy [30, 32]. Above all, hex-derived networks are dual of (3,12,12) and (4,8,8) carbon allotrope (nanocones, nanotubes), respectively. These networks may occur as a 2D molecular graph of some chemical compound from inorganic chemistry.The chemical industry produces dyes and polymers like plastics from organic compounds called benzenoid hydrocarbons, whose molecular graphs are hexagonal networks. Consequently, this class of compounds is a subject of great interest for theoretical chemists [33, 34]. We derive hex-derived networks (molecular structure of our interest) systematically from hexagonal structures. Now to our astonishment, these new structures represent inorganic nanostructures.Novel and valuable graph structures can be obtained from existing graphs by manipulating operations such as union, intersection, product, merger, contraction, and subdivision (expansion). A subdivisionSΓ of a graph Γ is obtained by replacing each edge of Γ with a path of length 2. Thus, the order and size of SΓ are VΓ+EΓ and 2EΓ. Moreover, SΓ is always bipartite, with two disjoint color classes EΓ and VΓ, respectively. All the original vertices of Γ preserve their degrees, while all the new vertices have degrees 2. Subdivision operation realizes lot of applications in accomplishment of various results in graph theory. Subdivision of planar graph is always planar. Two graphs are said to be homeomorphic if both can be obtained from certain graph by a sequence of edge subdivision. A graph homeomorphic to K3,3 (complete bipartite graph) or K5 (complete graph on five vertices) is called Kuratowski’s graph. A graph (finite) is planar if it does not contain Kuratowski’s graph. Figure 2 demonstrates a prominent nonplanar molecular graph [35] containing subdivision of K5.Figure 2
Subdivision ofK5 inside ferrocenophane molecular graph.The networks under our discussion, being subdivision ofHDN1n, HDN2n, and HDN3n, are fairly interesting, complex, and diversified in nature. We first compute and provide exact formulas of certain structural descriptors such as ABC, ABC4, GA, and GA5 of SHDN2n. In addition, we compute the first Zagreb, the second Zagreb, the modified Zagreb, augmented Zagreb, the inverse Randic′, the general Randic′, the symmetric division degree, the inverse sum, and the harmonic indices of SHDN2n and SHDN3n using latest technique of algebraic polynomials. Also, we perform a graphical analysis to compare networks SHDN1n, SHDN2n, and SHDN3n using the above-mentioned descriptors.The rest of paper is arranged as follows: Section II deals with the historic development and background of most imperative definitions, formulas their applications, and outline of the technique used. Section III comprises of derivation of main results relating to various structural descriptors of the subdivided hex-derived network. In the end, Section IV includes discussion and final remarks about results.


## 2. Preliminaries and Background

Harry Wiener (1947) was the pioneer who presents the idea of the distance-based molecular descriptor while studying the boiling point of hydrocarbons [5]. Initially, this descriptor got the name as path number and, eventually, recognized as Wiener index [36]. Gutman and Trinajstic′ (1972) originated the idea of two molecular descriptors to examine the total π-electron of hydrocarbons [37]. The formulas of the first Zagreb, the second Zagreb, and the modified Zagreb descriptors are given
(1)M1Γ=∑xy∈EΓdx+dy,M2Γ=∑xy∈EΓdx×dy,mM2Γ=∑xy∈EΓ1dxdy.Milan Randic′ (1975) proposed a significant descriptor known as branching/connectivity index [4]. It is very influential and is considered to be the prototype of degree-based indices. It measures the extent of branching in saturated hydrocarbons. Many descriptors are variations of this index and are defined by the formula
(2)R−12Γ=∑xy∈EΓ1dxdy.Böllöbás and Erdös [38] and Amic et al. [39], working independently in the same year (1998), broaden the concept of Randic′ connectivity index to all reals except zero, and the descriptor thus attained is called general Randic′ index and is defined as
(3)RβΓ=∑xy∈EΓ1dxdyβ.Formula of inverse Randic′ index is outlined as
(4)RRβΓ=∑xy∈EΓdxdyβ.Zhou and Trinajstic′ [40] introduced a variant of Randic′ index, and its additive version is called the sum-connectivity index (SCI). The formula for SCI is given by
(5)χ−12Γ=χ−12Γ=∑vw∈EΓ1dv+dw.Estrada et al. (1998) offered a significant descriptor termed as ABC index [9] and is used to describe the strain energy of cycloalkanes and the stability of alkanes.
(6)ABCΓ=∑xy∈EΓdx+dy−2dxdy.In 2009,Vukic˘evic′ et al. [11] introduced the so called geometric arithmetic index and is defined as
(7)GAΓ=∑xy∈EΓ2dxdydx+dy.Symmetric division degree (SDD) and inverse sum index (ISI) are good predictors of the total surface area of poly-chlorobiphenyls (bicyclic graphs) and octane isomers (tree graphs), respectively. These descriptors are defined as follows:(8)SDDΓ=∑xy∈EΓdx2+dy2dxdy,ISIΓ=∑xy∈EΓdxdydx+dy.Augmented Zagreb index (AZI) [41] correlates well with the heat of the formation of heptanes and octanes. The formula of AZI is described as
(9)AZIΓ=∑xy∈EΓdxdydx+dy−23.Another descriptor of our interest includes harmonic index (HI) and is defined by the formula(10)HIΓ=∑xy∈EΓ2dx+dy.Two prominent descriptors are denoted byABC4 and GA5 and are suggested by Ghorbani et al. [10] and Graovac et al. [12], respectively. Unlike other degree-based descriptors, the computation of ABC4 and GA5 involves the partitioning of edge set for neighbors’ degree sum of end vertices of each edge in the molecular graph. The formulas of ABC4 and GA5 are:
(11)ABC4Γ=∑xy∈EΓSx+Sy−2SxSy,(12)GA5Γ=∑xy∈EΓ2SxSySx+Sy.Motivated by current research work related to augmented Zagreb index, Hosamani [42] initiated the idea of Sanskruti index SΓ. The formula of the Sanskruti index is given as
(13)SΓ=∑xy∈EΓSxSySx+Sy−23.In [43–45], Ali et al. figured out extremal results and provided bounds on the harmonic index, augmented Zagreb index, and atom bond connectivity index, respectively. We suggest interested readers the comprehensive survey papers [46, 47] related to Randic′ index and geometric arithmetic index, respectively.Deutsch and Klavz^ar originated the concept of M-polynomial for graph Γ=VΓ,EΓ and are represented as [20]
(14)MΓ;a,b=fa,b=∑i≤jmijΓaibj,where mijΓ denote the number of edges ab∈EΓ provided daΓ,dbΓ=i,j. Certain prominent molecular descriptors are derived using M-polynomial and are presented in Table 1.Table 1
Formulas of certain significant structural descriptors based on M-polynomial.
Structural descriptorsfa,bFormulas derived from M-polynomialM1a+bDa+Dbfa,ba=b=1M2a.bDa.Dbfa,ba=b=1mM21/a.bSa.Sbfa,ba=b=1Rβa.bβDaβ.Dbβfa,ba=b=1RRβ1/a.bβSaβ.Sbβfa,ba=b=1SDDa2+b2/abDaSb+SaDbfa,ba=b=1HI2/a+b2SaJfa,ba=1ISIa.b/a+bSaJDaDbfa,ba=1AZIa.b/a+b−23Sa3Q−2JDa3Db3fa,ba=1whereDaf=a∂f/∂a,Dbf=b∂f/∂b,Saf=∫0aft,b/tdt,Sbf=∫0bfa,t/tdt,Jfa,b=fa,a,Qβfa,b=aβfa,b.In [48], Paul Manuel et al. conjectured about the weak minimum metric dimension (WMMD) bound on HDN1n and HDN2n. Xu and Fan [32] studied and partially answered this question by providing a refined WMMD bound. Shao et al. [49] worked out the metric dimension for HDN1n. Song et al. [50] and Imran et al. [51] discussed some topological properties and entropy-based descriptors of HDN1n and HDN2n, respectively. Wei et al. [52] and Ali et al. [53] computed degree-based and VE degree-based descriptors of the hex-derived network of the third kind HDN3n, respectively. Ahmad et al. [54] figured out certain topological properties of the subdivided hex-derived network of the first kind SHDN1n using M-polynomial. First, we intend to investigate topological descriptors of the subdivided hex-derived network of the second SHDN2n and the third kind SHDN3n. Then, we present a graphical comparative analysis based on these descriptors to propose a relation between physicochemical and structural properties of the networks SHDN1n, SHDN2n, and SHDN3n.


## 3. Main Results

### 3.1. Results for Subdivided Hex-Derived NetworksSHDN2n

The construction of the second type of hex-derived networkHDN2n is a bit complicated and is accomplished by merging the n-dimensional hexagonal network with (n-1)-dimensional honeycomb network. Alternatively, the architecture of HDN2n can be designed by first taking the union ofn-dimensional hexagonal network with its bounded dual ((n-1)-dimensional honeycomb network) and then by connecting every vertex of honeycomb with all the vertices of the corresponding triangle of hexagonal network [30]. For another architecture of HDN2n, see [32]. The order and size of hex-derived network HDN2n is 9n2−15n+7 and 36n2−2n+1, respectively [48]. The network of our interest SHDN2n can be attained from HDN2n by elementary subdivision. Without loss of generality, now to onward, suppose SHDN2n=Γ2n. Figure 3 depicts a subdivision of a 4-dimensional hex-derived network of the second kind.Theorem 1.
LetΓ2n denote the subdivided hex-derived network of second type, and then
(15)ABCΓ2n=1272n2−144n+72,GAΓ2n=726+1263n27+230107+1433−10867−273n+493+66−14143,ABC4Γ2n=3222+3464+30+46n2+26+1277+1219110−11222+6267−10303+1231266+1241456−10423−11464+67711+317017+1518291+344234+89713n+5425−26−1277+249110+522−18267−2431266+8303−2441456+8423+5462−127711−3018291−289713,GA5Γ2n=31265+2437+12n2+421107+63011+1709+123323+63925+64213−3065+1829+226611−6037+1211443−30n+815−21107−33011−123323−63925−34213+1265−1829−226611+2437+1211443.Proof.
SinceΓ2n is subdivision of HDN2n that is achieved by replacing each edge of HDN2n with a path of length 2, therefore, the number of vertices of Γ2n is VΓ2n=VHDN2n+EHDN2n=45n2−87n+43, and the number of edges of Γ2n is EΓ2n=2EHDN2n=72n2−2n+1. After analyzing the graphs of Γ2n for different values of n, we recognize four kinds of edges concerning degrees of end vertices of each edge that initiated the edge partitioning of Γ2n and are described in Table 2.Figure 3
A 4-dimensional subdivided hex-derived network of a secondΓ14.Table 2
Edge partitioning ofΓ2n with respect to degree of end vertices.
dx,dy:xy∈EΓ2n(2,5)(2,6)(2,7)(2,12)No. of edges30n36n2−3n+242n−2123n2−9n+7Using formulas presented by Equations (6) and (7), we can proceed with the computation of ABC and GA descriptors as given
(16)ABCΓ2n=∑xy∈EΓ2ndx+dy−2dxdy=510E1+612E2+714E3+1224E4=1230n+362n2−3n+2+422n−2+1223n2−9n+7=1272n2−144n+72,GAΓ2n=∑xy∈EΓ2n2dxdydx+dy=2107E1+2128E2+2149E3+22414E4=726+1263n27+230107+1433−10867−273n+493+66−14143.To get results forABC4Γ2n and GA5Γ2n, we have to partition edge set of Γ2n based on degree sum of neighbors of end vertices for all edges of Γ2n which is presented in Table 3:Table 3
Neighbors’ degree sum based edge partitioning ofΓ1n.
Sx,Sy:xy∈EΓ1Number of edgesSetsSx,Sy:xy∈EΓ1Number of edgesSets(10,10)36E1(12,18)63n2−10n+8E9(10,11)12n−2E2(13,14)6n−2E10(10,12)12n−1E3(14,14)12n−3E11(10,17)6nE4(14,19)12n−2E12(11,12)12n−2E5(17,24)6nE13(12,12)63n2−11n+10E6(18,24)63n2−10n+8E14(12,13)6n−2E7(19,24)12n−2E15(12,14)12n−1E8(24,24)63n2−11n+10E16Using information presented in Table3 and Equation (11), the fourth version of ABC can be computed as
(17)ABC4Γ2n∑xy∈EΓ2nSx+Sy−2SxSy+950E1+19110E2+16E3+534E4744E5+1172∣E6∣+23156∣E7∣+17∣E8∣+754∣E9∣+25182∣E10∣+1398∣E11∣+31266∣E12∣+13136∣E13∣+554∣E14∣+41456∣E15∣+23288∣E16∣=5425+1911012n−14+26n−1+317017n+2223n2−11n+10+67711n−2+89713n−2+1277n−1+1518291n−2+4233n2−10n+8+6267n−3+1231266n−2+3033n2−10n+8+344234n+1241456n−2+4643n2−11n+10=3222+3464+30+46n2+26+1277+1219110−11222+6267−10303+1231266−10423−11464+67711+317017+1518291+344234+1241456+89713n+5425−26−1277+249110+522−18267−2431266+8303−2441456+8423+5462−127711−3018291−289713.Likewise using the information provided in Table3 for GA5, we have
(18)GA5Γ2n=∑xy∈EΓ2n2SxSySx+Sy=10010∣E1∣+211021∣E2∣+12011∣E3∣+217027∣E4∣+13223∣E5∣+214412∣E6∣+215625∣E7∣+16813∣E8∣+21615∣E9∣+218227∣E10∣+19614∣E11∣+226633∣E12∣+240841∣E13∣+43221∣E14∣+245643∣E15∣+57624∣E16∣=36+81107n−2+243011n−1+41709n+483323n−2+63n2−11n+10+243925n−2+244213n−1+12653n2−10n+8+41829n−2+12n−3+826611n−2+24373n2−10n+8+2410241n+4811443n−2+63n2−11n+10GA5Γ2n=312652437+12n2+421107+63011+1709+123323+63925+64213−3065+1829+226611−6037+1211443−30n+815−21107−33011−12332363925−34213+1265−1829−226611+2437+1211443.Theorem 2.
LetΓ2ndenote the subdivided hex-derived network of a second kind, and then, the M-polynomial ofΓ2nis given by(19)MΓ2n;a,b=30na2b5+36n2−108n+72a2b6+42n−84a2b7+36n2−108n+84a2b12.Proof.
Data provided in Table2 can further be rewritten as
(20)E1Γ2n=xy∈EΓ2n:dx=2,dy=5,∣E1Γ2n∣=30n,E2Γ2n=xy∈EΓ2n:dx=2,dy=6,∣E2Γ2n∣=36n2−108n+72,E3Γ2n=xy∈EΓ2n:dx=2,dy=7,∣E3Γ2n∣=42n−84,E4Γ2n=xy∈EΓ2n:dx=2,dy=12,∣E4Γ2n∣=36n2−108n+84.
Using formula of M-polynomial presented by Equation (14), we proceed as
(21)MΓ2n;a,b=∑i≤jmijaibj∑2≤5m25a2b5+∑2≤6m26a2b6+∑2≤7m27a2b7+∑2≤12m212a2b12=E1Γ2na2b5+E2Γ2na2b6+E3Γ2na2b7+E4Γ2na2b12=30na2b5+36n2−108n+72a2b6+42n−84a2b7+36n2−108n+84a2b12Proposition 3.
SupposeΓ2nrepresent subdivided hex-derived network of a second kind, and then the first Zagreb, the second Zagreb, the modified Zagreb, the augmented Zagreb, the inverse Randic′, the general Randic′, the symmetric division degree, the inverse sum, and the harmonic indices are(1)
M1Γ2n=1266n2−149n+83,(2)
M2Γ2n=2454n2−125n+71,(3)
mM2Γ2n=1/29n2−9n+7,(4)
RβΓ2n=3×12β+11+2βn2+6×2β5β+1+7β+1−3×6β+1−22β+1×3β+2n+12×2β6β+1−7β+1+7×12β,(5)
RRβΓ2n=3121−β+12×24−βn2+3101−β+141−β−3×121−β−36×24−βn+6121−β−141−β+14×24−β,(6)
SSDΓ2n=2171n2−390n+220,(7)
HIΓ2n=1/21297n2−515n+238,(8)
ISIΓ2n=2/211215n2−1213n+1274,(9)
AZIΓ2n=576n2−2n+1.Proof.
Consider equation of M-polynomial derived in Theorem2(22)MΓ;a,b=30na2b5+36n2−108n+72a2b6+42n−84a2b7+36n2−108n+84a2b12.
Now by applying a combination of the differential and integral operators provided in the derivation Table2 on M-polynomial, we get
(23)Da+Dbfa,b210na2b5+288n2−3n+2a2b6+378n−2a2b7+1683n2−9n+7a2b12DaDbfa,b300na2b5+432n2−3n+2a2b6+588n−2a2b7+2883n2−9n+7a2b12.SaSbfa,b=3na2b5+3n2−3n+2a2b6+3n−2a2b7+123n2−9n+7a2b12.DaβDbβfa,b=10β+13na2b5+12β+13n2−9n+6a2b6+14β+13n−6a2b7+24β36n2−108n+84a2b12.SaJfa,b=307na7+9n2−3n+2a8+14n−283a9+673n2−9n+7a14SaJDaDbfa,b=3007na7+54n2−3n+2a8+1963n−2a9+14473n2−9n+7a14.Sa3Q−2JDa3Db3fa,b=240na5+288n2−3n+2a6+336n−2a7+963n2−9n+7a12.Now, by applying formulas of molecular descriptors derived from M-polynomial depicted in Table 11, we get
(1)
M1Γ2n=Da+Dbfa,ba=b=1=1266n2−149n+83,(2)
M2Γ2n=DbDafa,ba=b=1=2454n2−125n+71,(3)
mM2Γ2n=SaSbfa,ba=b=1=1/29n2−9n+7,(4)
RβΓ2n=DaβDbβfa,ba=b=1=3×12β+11+2βn2+6×2β5β+1+7β+1−3×6β+1−22β+1×3β+2n+12×2β6β+1−7β+1+7×12β,(5)
RRβΓ2n=SaβSbβfa,ba=b=1=3121−β+12×24−βn2+3101−β+141−β−3×121−β−36×24−βn+6121−β−141−β+14×24−β,(6)
SSDΓ2n=DaSb+SaDbfa,ba=b=1=2171n2−390n+220,(7)
HIΓ2n=2SaJfa,aa=1=1/21297n2−515n+238,(8)
ISIΓ2n=SaJDaDbfa,ba=b=1=2/211215n2−1213n+1274,(9)
AZIΓ2n=Sa3Q−2JDa3Db3fa,ba=b=1=576n2−2n+1.


### 3.2. Results for Subdivided Hex-Derived NetworksSHDN3n

Simonraj et al. [55] proposed a new variant of the hex-derived network called the third kind of hex-derived network HDN3n and computed the metric dimension for HDN3n. The construction of network HDN3n is quite simple and accomplished by replacing each triangle in the hexagonal mesh HXn with an octahedral graph. The order and size of HDN3n are 21n2−39n+19 and 63n2−123n+60, respectively. The network of our concern SHDN3n is fairly interesting and can be obtained from HDN3n by replacing each edge with a path of length 2. For the sake of simplicity, throughout, let SHDN3n=Γ3n. Figure 4 depicts a subdivision of a 4-dimensional hex-derived network of the third kind.Theorem 4.
LetΓ3nrepresent the subdivided hex-derived network of a third kindSHDN3n,and then, the following exact formulas hold(24)ABCΓ3n=12126n2−246n+120,GAΓ3n=2402+162n25−24802−1005+486n5+1157202−6005+14014+1134,ABC4Γ3n=9914+6077+1170n222−2079014−1069235+1890077−138638−1981610−330434+42370n2310+1392701606514+21420374−3304835+127577+61203542+19806902+3366518+13860119−7068638−67321610+785459−11220434+6545070,GA5Γ3n=139211+108022+7830n2145−156182411−9539557−22968070−10846035+121176022+6832980n54230+962219−288711+112115+4815425+2423831−1447017+201413+488537−435+5042229−97+402061.Proof.
We know that the subdivided networkΓ3n is obtained from HDN3n by replacing each edge with a path of length 2; hence, a total number of vertices of Γ3n are ∣VΓ3n∣=∣VHDN3n∣+∣EHDN3n∣=21n2−39n+19+63n2−123n+60=84n2−162n+79, and number of edges of Γ3n are ∣EΓ3n∣=2∣EHDN3n∣=126n2−246n+120. We observe and recognize four types of edges with respect to degree of end vertices of each edge. This provides basis for edge partitioning of Γ3n and is outlined in Table 4.Figure 4
A 4-dimensional subdivided hex-derived network of a third kindΓ34.Table 4
Edge partitioning ofΓ3n with respect to degree of end vertices.
dx,dy:xy∈EΓ3nNo. of edgesSets(2,4)72n2−144n+72E1(2,7)42E2(2,10)60n−2E3(2,18)54n2−162n+126E4Using formulas presented by Equations (6) and (7), we can proceed with the computation of ABC and GA descriptors as
(25)ABCΓ3n=∑xy∈EΓ3ndx+dy−2dxdy=48E1+714E2+1020E3+1836E4=1272n2−144n+72+1242+1260n−120+1254n2−162n+126=12126n2−246n+120,GAΓ3n=∑xy∈EΓ3n2dxdydx+dy=286∣E1∣+2149∣E2∣+22012∣E3∣+23620∣E4∣=8372n2−144n+72+28143+1020n−2+3554n2−162n+126=2402+162n25−24802−1005+486+1157202−6005+14014+1134.In order to findABC4Γ3n and GA5Γ3n, we need edge partition of Γ3n based on degree sum of neighbors of end vertices of each edge of Γ3 which is presented in Table 5.Table 5
Neighbors’ degree sum based edge partitioning ofΓ3n.
Sx,Sy:Number of edgesSetsSx,Sy:Number of edgesSetsxy∈EΓ3nxy∈EΓ3n(8,8)36n2−72n+36E1(17,20)12E9(8,11)24E2(20,20)12n−3E10(8,14)36n−2E3(20,28)12n−2E11(8,22)36n2−108n+84E4(22,36)36n2−108n+84E12(11,14)24E5(25,36)6E13(14,17)12E6(28,36)12n−2E14(14,20)36n−2E7(36,36)18n2−66n+60E15(14,25)6E8Using information presented in Table5 and Equation (11), we proceed for closed form formula of ABC4 as
(26)ABC4Γ3n=∑xy∈EΓ3nSx+Sy−2SxSy=1464∣E1∣+1788∣E2∣+20112∣E3∣+28176∣E4∣+23154∣E5∣+29238∣E6∣+32280∣E7∣+37350∣E8∣+35240∣E9∣+38400∣E10∣+46560∣E11∣+56792∣E12∣+59900∣E13∣+621008∣E14∣+701296∣E15∣=14836n2−72n+36+2437444+351436n−72+772236n2−108n+84++2423154+1229238+3228036n−72+637350+1211934+382012n−36+773336n2−108n+84+161014012n−24+65930+4348412n−24+703618n2−66n+60=9914+6077+1170n222−2079014−330434−1069235+1890077−138638−1981610+42370n2310+1392701606514+21420374−3304835+127577+61203542+19806902+3366518+13860119−7068638−67321610+785459−11220434+6545070.Similarly, by employing formula forGA5 along with data provide in Table 5, we proceed as
(27)GA5Γ3n=∑xy∈EΓ3n2SxSySx+Sy=26416E1+28819E2+211222E3+217630E4+215425∣E5∣+223831E6+228034E7+235039E8+234037E9+240040E10+256048E11+279258E12+290061E13+2100864E14+2129672E15=36n2−72n+36+488819+1121136n−72+1761536n2−108n+84+4815425+2423831+2801736n−72+1235037+2434037+12n−36+5602412n−24+7922936n2−108n+84+36061+10083212n−24+18n2−66n+60GA5Γ3n=139211+108022+7830n2145−156182411−9539557−22968070−10846035+121176022+6832980n54230+962219−288711+112115+4815425+2423831−1447017+201413+488537−435+5042229−97+402061.Theorem 5.
LetΓ3nrepresent the subdivided hex-derived network of the third kind, and then the formula of M-polynomial forΓ3nis given as(28)MΓ3n;a,b=72n2−144n+72a2b4+42a2b7+60n−1a2b10+54n2−162n+126a2b18.Proof.
Edge partition described in Table4 can further be simplified as
(29)E1Γ3n=xy∈EΓ3n:dx=2,dy=4,∣E1Γ3n∣=72n2−144n+72.E2Γ3n=xy∈EΓ3n:dx=2,dy=7,∣E2Γ3n∣=42.E3Γ3n=xy∈EΓ3n:dx=2,dy=10,∣E3Γ3n∣=60n−2.E4Γ3n=xy∈EΓ3n:dx=2,dy=18,E4Γ3n∣=54n2−162n+126.
Using formula of M-polynomial presented by Equation (14), we proceed as
(30)MΓ3n;a,b=∑i≤jmijaibj+∑2≤18m218a2b18=∑2≤4m24a2b4+∑2≤7m27a2b7+∑2≤10m210a2b10=E1Γ3na2b4+E2Γ3na2b7+E3Γ3na2b10+E4Γ3na2b18=72n2−144n+72a2b4+42a2b7+60n−1a2b10+54n2−162n+126a2b18.Proposition 6.
SupposeΓ3ndenote subdivided hex-derived network of the third kind, and then, the first Zagreb, the second Zagreb, the modified Zagreb, the augmented Zagreb, the inverse Randic′, the general Randic′, the symmetric division degree, the inverse sum, and the harmonic indices are(1)
M1Γ3n=1512n2−3384n+1890,(2)
M2Γ3n=2520n2−5784n+3300,(3)
mM2Γ3n=1/221n2−39n+19,(4)
RβΓ3n=981+β+61+2βn2−36×81+β−201+β+9×61+2βn+33×81+β−2×201+β+141+β+7×61+2β,(5)
RRβΓ3n981−β+61−2βn2−36×81−β−201−β+9×61−2βn+33×81−β−2×201−β+141−β+7×61−2β,(6)
SSDΓ3n=600n2−1380n+791,(7)
HIΓ3n=1/15441n2−813n+3897,(8)
ISIΓ3n=1260n2−2892n+1650,(9)
AZIΓ3n=1008n2−1968n+960.Proof.
Let(31)MΓ3n;a,b=72n2−144n+72a2b4+42a2b7+60n−1a2b10+54n2−162n+126a2b18.
Now by applying a combination of the differential and integral operators provided in the derivation Table4 on M-polynomial, we get
(32)Da+Dbfa,b=672n2−144n+72a2b4+378a2b7+1260n−120a2b10+2054n2−162n+126a2b18.DaDbfa,b=872n2−144n+72a2b4+588a2b7+2060n−120a2b10+3654n2−162n+126a2b18.SaSbfa,b=9n2−18n+9a2b4+3a2b7+3n−6a2b10+123n2−9n+7a2b18.DaβDbβfa,b=8β72n2−144n+72a2b4+3×141+βa2b7+20β60n−120a2b10+36β54n2−162n+126a2b18.SaβSbβfa,b=81−β9n2−18n+9a2b4+3×141−βa2b7+201−β3n−6a2b10+61−2β9n2−27n+21a2b18.SbDa+SaDbfa,b336n2−72n+36a2b4+159a2b7+2612n−24a2b10+826n2−18n+14a2b18.Jfa,b=fa,a=72n2−2n+1a6+42a9+60n−2a12+183n2−9n+7a20.SaJfa,b=12n2−2n+1a6+143a9+5n−2a12+9103n2−9n+7a20SaJDaDbfa,b=96n2−2n+1a6+1963a9+100n−2a12+9554n2−162n+126a20.Sa3Q−2JDa3Db3fa,b=576n2−2n+1a4+336a7+480n−2a10+1443n2−9n+7a20.Now, by applying formulas of molecular descriptors derived from M-polynomial depicted in table??, we get:(1)
M1Γ3n=Da+Dbfa,ba=b=1=1512n2−3384n+1890,(2)
M2Γ3n=DbDafa,ba=b=1=2520n2−5784n+3300,(3)
mM2Γ3n=SaSbfa,ba=b=1=1/221n2−39n+19,(4)
RβΓ3n=DaβDbβfa,ba=b=1981+β+61+2βn2−36×81+β−201+β+9×61+2βn+33×81+β−2×201+β+141+β+7×61+2β,(5)
RRβΓ3n=SaβSbβfa,ba=b=1=981−β+61−2βn2−36×81−β−201−β+9×61−2βn+33×81−β−2×201−β+141−β+7×61−2β,(6)
SSDΓ3n=DaSb+SaDbf(a,b)a=b=1=600n2−1380n+791,(7)
HIΓ3n=2SaJfa,aa=1=1/15441n2−813n+3897,(8)
ISIΓ3n=SaJDaDbfa,ba=b=1=1260n2−2892n+1650,(9)
AZIΓ3n=Sa3Q−2JDa3Db3fa,ba=b=1=1008n2−1968n+960.


### 3.3. Results for Subdivided Hex-Derived NetworkSHDN1n.

In an earlier article [54], we studied and figured out numerous topological invariants of SHDN1n=Γ1n. A 4-dimensional subdivided hex-derived network of first kind Γ14 is shown in Figure 5. To perform a comparative analysis, the aforementioned results are presented here.Theorem 7 (see [54]).
LetΓ1ndenote the subdivided hex-derived network of first kind, and then(33)ABCΓ1n=1254n2−102n+48,GAΓ1n=612635n2+410524514−3786−40524n+41051896+22510−49014+31524,ABC4Γ1n=31905+37010+464n2+6267+1231266+641114−11464−91905−937010+977035n+6+26+125+127−18267−2431266−1241114+5462+317017+71905+32105−737010+344234−1877035,GA5Γ1n=33241091+6n2+9152−29161091+335+4811443+826611−54n+4837+1653+3241013−915+243011−635+244213−2410241−9611443+41709−1626611+24.Proposition 8 (see [54]).
SupposeΓ1ndenote subdivided hex-derived network of first type, and then, the first Zagreb, the second Zagreb, the modified Zagreb, the augmented Zagreb, the inverse Randic′, the general Randic′, the symmetric division degree, the inverse sum, and the harmonic indices are(1)
M1Γ1n=1833n2−73n+40,(2)
M2Γ1n=1281n2−185n+107,(3)
mM2Γ1n=1/29n2−15n+7,(4)
RβΓ1n=3×6β+11+22β+1n2+6×2β7β+1−2×3β+1−22β+1×3β+2η+310β+1+6β+1−2×14β+1+28×24β,(5)
RRβΓ1n=361−β+12×24−βn2+3141−β−2×61−β−36×24−βη+3101−β+61−β−2×141−β+28×24−β,(6)
SDDΓ1n=261n2−585n+326,(7)
HIΓ1n=4/105324n2−538n+239,(8)
ISIΓ1n=4/1052187n2−4279n+2042,(9)
AZIΓ1n=489n2−17n+8.Proposition 9.
LetΓ1n, Γ2n, andΓ3nbe the the hex-derived networks of type 1, type 2, and type 3, respectively. Then(34)χ−12Γ1n183575+514n2+235245−1265−13514n+235635+757+10514−490,χ−12Γ2n17632+1814n2−1892−307+5414−98n+1262+4214−196,χ−12Γ3n15606+275n2−1206−503+815n+606−1003+635+70.Figure 5
A 4-dimensional subdivided hex-derived network of a first kindΓ14.


## 3.1. Results for Subdivided Hex-Derived NetworksSHDN2n

The construction of the second type of hex-derived networkHDN2n is a bit complicated and is accomplished by merging the n-dimensional hexagonal network with (n-1)-dimensional honeycomb network. Alternatively, the architecture of HDN2n can be designed by first taking the union ofn-dimensional hexagonal network with its bounded dual ((n-1)-dimensional honeycomb network) and then by connecting every vertex of honeycomb with all the vertices of the corresponding triangle of hexagonal network [30]. For another architecture of HDN2n, see [32]. The order and size of hex-derived network HDN2n is 9n2−15n+7 and 36n2−2n+1, respectively [48]. The network of our interest SHDN2n can be attained from HDN2n by elementary subdivision. Without loss of generality, now to onward, suppose SHDN2n=Γ2n. Figure 3 depicts a subdivision of a 4-dimensional hex-derived network of the second kind.Theorem 1.
LetΓ2n denote the subdivided hex-derived network of second type, and then
(15)ABCΓ2n=1272n2−144n+72,GAΓ2n=726+1263n27+230107+1433−10867−273n+493+66−14143,ABC4Γ2n=3222+3464+30+46n2+26+1277+1219110−11222+6267−10303+1231266+1241456−10423−11464+67711+317017+1518291+344234+89713n+5425−26−1277+249110+522−18267−2431266+8303−2441456+8423+5462−127711−3018291−289713,GA5Γ2n=31265+2437+12n2+421107+63011+1709+123323+63925+64213−3065+1829+226611−6037+1211443−30n+815−21107−33011−123323−63925−34213+1265−1829−226611+2437+1211443.Proof.
SinceΓ2n is subdivision of HDN2n that is achieved by replacing each edge of HDN2n with a path of length 2, therefore, the number of vertices of Γ2n is VΓ2n=VHDN2n+EHDN2n=45n2−87n+43, and the number of edges of Γ2n is EΓ2n=2EHDN2n=72n2−2n+1. After analyzing the graphs of Γ2n for different values of n, we recognize four kinds of edges concerning degrees of end vertices of each edge that initiated the edge partitioning of Γ2n and are described in Table 2.Figure 3
A 4-dimensional subdivided hex-derived network of a secondΓ14.Table 2
Edge partitioning ofΓ2n with respect to degree of end vertices.
dx,dy:xy∈EΓ2n(2,5)(2,6)(2,7)(2,12)No. of edges30n36n2−3n+242n−2123n2−9n+7Using formulas presented by Equations (6) and (7), we can proceed with the computation of ABC and GA descriptors as given
(16)ABCΓ2n=∑xy∈EΓ2ndx+dy−2dxdy=510E1+612E2+714E3+1224E4=1230n+362n2−3n+2+422n−2+1223n2−9n+7=1272n2−144n+72,GAΓ2n=∑xy∈EΓ2n2dxdydx+dy=2107E1+2128E2+2149E3+22414E4=726+1263n27+230107+1433−10867−273n+493+66−14143.To get results forABC4Γ2n and GA5Γ2n, we have to partition edge set of Γ2n based on degree sum of neighbors of end vertices for all edges of Γ2n which is presented in Table 3:Table 3
Neighbors’ degree sum based edge partitioning ofΓ1n.
Sx,Sy:xy∈EΓ1Number of edgesSetsSx,Sy:xy∈EΓ1Number of edgesSets(10,10)36E1(12,18)63n2−10n+8E9(10,11)12n−2E2(13,14)6n−2E10(10,12)12n−1E3(14,14)12n−3E11(10,17)6nE4(14,19)12n−2E12(11,12)12n−2E5(17,24)6nE13(12,12)63n2−11n+10E6(18,24)63n2−10n+8E14(12,13)6n−2E7(19,24)12n−2E15(12,14)12n−1E8(24,24)63n2−11n+10E16Using information presented in Table3 and Equation (11), the fourth version of ABC can be computed as
(17)ABC4Γ2n∑xy∈EΓ2nSx+Sy−2SxSy+950E1+19110E2+16E3+534E4744E5+1172∣E6∣+23156∣E7∣+17∣E8∣+754∣E9∣+25182∣E10∣+1398∣E11∣+31266∣E12∣+13136∣E13∣+554∣E14∣+41456∣E15∣+23288∣E16∣=5425+1911012n−14+26n−1+317017n+2223n2−11n+10+67711n−2+89713n−2+1277n−1+1518291n−2+4233n2−10n+8+6267n−3+1231266n−2+3033n2−10n+8+344234n+1241456n−2+4643n2−11n+10=3222+3464+30+46n2+26+1277+1219110−11222+6267−10303+1231266−10423−11464+67711+317017+1518291+344234+1241456+89713n+5425−26−1277+249110+522−18267−2431266+8303−2441456+8423+5462−127711−3018291−289713.Likewise using the information provided in Table3 for GA5, we have
(18)GA5Γ2n=∑xy∈EΓ2n2SxSySx+Sy=10010∣E1∣+211021∣E2∣+12011∣E3∣+217027∣E4∣+13223∣E5∣+214412∣E6∣+215625∣E7∣+16813∣E8∣+21615∣E9∣+218227∣E10∣+19614∣E11∣+226633∣E12∣+240841∣E13∣+43221∣E14∣+245643∣E15∣+57624∣E16∣=36+81107n−2+243011n−1+41709n+483323n−2+63n2−11n+10+243925n−2+244213n−1+12653n2−10n+8+41829n−2+12n−3+826611n−2+24373n2−10n+8+2410241n+4811443n−2+63n2−11n+10GA5Γ2n=312652437+12n2+421107+63011+1709+123323+63925+64213−3065+1829+226611−6037+1211443−30n+815−21107−33011−12332363925−34213+1265−1829−226611+2437+1211443.Theorem 2.
LetΓ2ndenote the subdivided hex-derived network of a second kind, and then, the M-polynomial ofΓ2nis given by(19)MΓ2n;a,b=30na2b5+36n2−108n+72a2b6+42n−84a2b7+36n2−108n+84a2b12.Proof.
Data provided in Table2 can further be rewritten as
(20)E1Γ2n=xy∈EΓ2n:dx=2,dy=5,∣E1Γ2n∣=30n,E2Γ2n=xy∈EΓ2n:dx=2,dy=6,∣E2Γ2n∣=36n2−108n+72,E3Γ2n=xy∈EΓ2n:dx=2,dy=7,∣E3Γ2n∣=42n−84,E4Γ2n=xy∈EΓ2n:dx=2,dy=12,∣E4Γ2n∣=36n2−108n+84.
Using formula of M-polynomial presented by Equation (14), we proceed as
(21)MΓ2n;a,b=∑i≤jmijaibj∑2≤5m25a2b5+∑2≤6m26a2b6+∑2≤7m27a2b7+∑2≤12m212a2b12=E1Γ2na2b5+E2Γ2na2b6+E3Γ2na2b7+E4Γ2na2b12=30na2b5+36n2−108n+72a2b6+42n−84a2b7+36n2−108n+84a2b12Proposition 3.
SupposeΓ2nrepresent subdivided hex-derived network of a second kind, and then the first Zagreb, the second Zagreb, the modified Zagreb, the augmented Zagreb, the inverse Randic′, the general Randic′, the symmetric division degree, the inverse sum, and the harmonic indices are(1)
M1Γ2n=1266n2−149n+83,(2)
M2Γ2n=2454n2−125n+71,(3)
mM2Γ2n=1/29n2−9n+7,(4)
RβΓ2n=3×12β+11+2βn2+6×2β5β+1+7β+1−3×6β+1−22β+1×3β+2n+12×2β6β+1−7β+1+7×12β,(5)
RRβΓ2n=3121−β+12×24−βn2+3101−β+141−β−3×121−β−36×24−βn+6121−β−141−β+14×24−β,(6)
SSDΓ2n=2171n2−390n+220,(7)
HIΓ2n=1/21297n2−515n+238,(8)
ISIΓ2n=2/211215n2−1213n+1274,(9)
AZIΓ2n=576n2−2n+1.Proof.
Consider equation of M-polynomial derived in Theorem2(22)MΓ;a,b=30na2b5+36n2−108n+72a2b6+42n−84a2b7+36n2−108n+84a2b12.
Now by applying a combination of the differential and integral operators provided in the derivation Table2 on M-polynomial, we get
(23)Da+Dbfa,b210na2b5+288n2−3n+2a2b6+378n−2a2b7+1683n2−9n+7a2b12DaDbfa,b300na2b5+432n2−3n+2a2b6+588n−2a2b7+2883n2−9n+7a2b12.SaSbfa,b=3na2b5+3n2−3n+2a2b6+3n−2a2b7+123n2−9n+7a2b12.DaβDbβfa,b=10β+13na2b5+12β+13n2−9n+6a2b6+14β+13n−6a2b7+24β36n2−108n+84a2b12.SaJfa,b=307na7+9n2−3n+2a8+14n−283a9+673n2−9n+7a14SaJDaDbfa,b=3007na7+54n2−3n+2a8+1963n−2a9+14473n2−9n+7a14.Sa3Q−2JDa3Db3fa,b=240na5+288n2−3n+2a6+336n−2a7+963n2−9n+7a12.Now, by applying formulas of molecular descriptors derived from M-polynomial depicted in Table 11, we get
(1)
M1Γ2n=Da+Dbfa,ba=b=1=1266n2−149n+83,(2)
M2Γ2n=DbDafa,ba=b=1=2454n2−125n+71,(3)
mM2Γ2n=SaSbfa,ba=b=1=1/29n2−9n+7,(4)
RβΓ2n=DaβDbβfa,ba=b=1=3×12β+11+2βn2+6×2β5β+1+7β+1−3×6β+1−22β+1×3β+2n+12×2β6β+1−7β+1+7×12β,(5)
RRβΓ2n=SaβSbβfa,ba=b=1=3121−β+12×24−βn2+3101−β+141−β−3×121−β−36×24−βn+6121−β−141−β+14×24−β,(6)
SSDΓ2n=DaSb+SaDbfa,ba=b=1=2171n2−390n+220,(7)
HIΓ2n=2SaJfa,aa=1=1/21297n2−515n+238,(8)
ISIΓ2n=SaJDaDbfa,ba=b=1=2/211215n2−1213n+1274,(9)
AZIΓ2n=Sa3Q−2JDa3Db3fa,ba=b=1=576n2−2n+1.


## 3.2. Results for Subdivided Hex-Derived NetworksSHDN3n

Simonraj et al. [55] proposed a new variant of the hex-derived network called the third kind of hex-derived network HDN3n and computed the metric dimension for HDN3n. The construction of network HDN3n is quite simple and accomplished by replacing each triangle in the hexagonal mesh HXn with an octahedral graph. The order and size of HDN3n are 21n2−39n+19 and 63n2−123n+60, respectively. The network of our concern SHDN3n is fairly interesting and can be obtained from HDN3n by replacing each edge with a path of length 2. For the sake of simplicity, throughout, let SHDN3n=Γ3n. Figure 4 depicts a subdivision of a 4-dimensional hex-derived network of the third kind.Theorem 4.
LetΓ3nrepresent the subdivided hex-derived network of a third kindSHDN3n,and then, the following exact formulas hold(24)ABCΓ3n=12126n2−246n+120,GAΓ3n=2402+162n25−24802−1005+486n5+1157202−6005+14014+1134,ABC4Γ3n=9914+6077+1170n222−2079014−1069235+1890077−138638−1981610−330434+42370n2310+1392701606514+21420374−3304835+127577+61203542+19806902+3366518+13860119−7068638−67321610+785459−11220434+6545070,GA5Γ3n=139211+108022+7830n2145−156182411−9539557−22968070−10846035+121176022+6832980n54230+962219−288711+112115+4815425+2423831−1447017+201413+488537−435+5042229−97+402061.Proof.
We know that the subdivided networkΓ3n is obtained from HDN3n by replacing each edge with a path of length 2; hence, a total number of vertices of Γ3n are ∣VΓ3n∣=∣VHDN3n∣+∣EHDN3n∣=21n2−39n+19+63n2−123n+60=84n2−162n+79, and number of edges of Γ3n are ∣EΓ3n∣=2∣EHDN3n∣=126n2−246n+120. We observe and recognize four types of edges with respect to degree of end vertices of each edge. This provides basis for edge partitioning of Γ3n and is outlined in Table 4.Figure 4
A 4-dimensional subdivided hex-derived network of a third kindΓ34.Table 4
Edge partitioning ofΓ3n with respect to degree of end vertices.
dx,dy:xy∈EΓ3nNo. of edgesSets(2,4)72n2−144n+72E1(2,7)42E2(2,10)60n−2E3(2,18)54n2−162n+126E4Using formulas presented by Equations (6) and (7), we can proceed with the computation of ABC and GA descriptors as
(25)ABCΓ3n=∑xy∈EΓ3ndx+dy−2dxdy=48E1+714E2+1020E3+1836E4=1272n2−144n+72+1242+1260n−120+1254n2−162n+126=12126n2−246n+120,GAΓ3n=∑xy∈EΓ3n2dxdydx+dy=286∣E1∣+2149∣E2∣+22012∣E3∣+23620∣E4∣=8372n2−144n+72+28143+1020n−2+3554n2−162n+126=2402+162n25−24802−1005+486+1157202−6005+14014+1134.In order to findABC4Γ3n and GA5Γ3n, we need edge partition of Γ3n based on degree sum of neighbors of end vertices of each edge of Γ3 which is presented in Table 5.Table 5
Neighbors’ degree sum based edge partitioning ofΓ3n.
Sx,Sy:Number of edgesSetsSx,Sy:Number of edgesSetsxy∈EΓ3nxy∈EΓ3n(8,8)36n2−72n+36E1(17,20)12E9(8,11)24E2(20,20)12n−3E10(8,14)36n−2E3(20,28)12n−2E11(8,22)36n2−108n+84E4(22,36)36n2−108n+84E12(11,14)24E5(25,36)6E13(14,17)12E6(28,36)12n−2E14(14,20)36n−2E7(36,36)18n2−66n+60E15(14,25)6E8Using information presented in Table5 and Equation (11), we proceed for closed form formula of ABC4 as
(26)ABC4Γ3n=∑xy∈EΓ3nSx+Sy−2SxSy=1464∣E1∣+1788∣E2∣+20112∣E3∣+28176∣E4∣+23154∣E5∣+29238∣E6∣+32280∣E7∣+37350∣E8∣+35240∣E9∣+38400∣E10∣+46560∣E11∣+56792∣E12∣+59900∣E13∣+621008∣E14∣+701296∣E15∣=14836n2−72n+36+2437444+351436n−72+772236n2−108n+84++2423154+1229238+3228036n−72+637350+1211934+382012n−36+773336n2−108n+84+161014012n−24+65930+4348412n−24+703618n2−66n+60=9914+6077+1170n222−2079014−330434−1069235+1890077−138638−1981610+42370n2310+1392701606514+21420374−3304835+127577+61203542+19806902+3366518+13860119−7068638−67321610+785459−11220434+6545070.Similarly, by employing formula forGA5 along with data provide in Table 5, we proceed as
(27)GA5Γ3n=∑xy∈EΓ3n2SxSySx+Sy=26416E1+28819E2+211222E3+217630E4+215425∣E5∣+223831E6+228034E7+235039E8+234037E9+240040E10+256048E11+279258E12+290061E13+2100864E14+2129672E15=36n2−72n+36+488819+1121136n−72+1761536n2−108n+84+4815425+2423831+2801736n−72+1235037+2434037+12n−36+5602412n−24+7922936n2−108n+84+36061+10083212n−24+18n2−66n+60GA5Γ3n=139211+108022+7830n2145−156182411−9539557−22968070−10846035+121176022+6832980n54230+962219−288711+112115+4815425+2423831−1447017+201413+488537−435+5042229−97+402061.Theorem 5.
LetΓ3nrepresent the subdivided hex-derived network of the third kind, and then the formula of M-polynomial forΓ3nis given as(28)MΓ3n;a,b=72n2−144n+72a2b4+42a2b7+60n−1a2b10+54n2−162n+126a2b18.Proof.
Edge partition described in Table4 can further be simplified as
(29)E1Γ3n=xy∈EΓ3n:dx=2,dy=4,∣E1Γ3n∣=72n2−144n+72.E2Γ3n=xy∈EΓ3n:dx=2,dy=7,∣E2Γ3n∣=42.E3Γ3n=xy∈EΓ3n:dx=2,dy=10,∣E3Γ3n∣=60n−2.E4Γ3n=xy∈EΓ3n:dx=2,dy=18,E4Γ3n∣=54n2−162n+126.
Using formula of M-polynomial presented by Equation (14), we proceed as
(30)MΓ3n;a,b=∑i≤jmijaibj+∑2≤18m218a2b18=∑2≤4m24a2b4+∑2≤7m27a2b7+∑2≤10m210a2b10=E1Γ3na2b4+E2Γ3na2b7+E3Γ3na2b10+E4Γ3na2b18=72n2−144n+72a2b4+42a2b7+60n−1a2b10+54n2−162n+126a2b18.Proposition 6.
SupposeΓ3ndenote subdivided hex-derived network of the third kind, and then, the first Zagreb, the second Zagreb, the modified Zagreb, the augmented Zagreb, the inverse Randic′, the general Randic′, the symmetric division degree, the inverse sum, and the harmonic indices are(1)
M1Γ3n=1512n2−3384n+1890,(2)
M2Γ3n=2520n2−5784n+3300,(3)
mM2Γ3n=1/221n2−39n+19,(4)
RβΓ3n=981+β+61+2βn2−36×81+β−201+β+9×61+2βn+33×81+β−2×201+β+141+β+7×61+2β,(5)
RRβΓ3n981−β+61−2βn2−36×81−β−201−β+9×61−2βn+33×81−β−2×201−β+141−β+7×61−2β,(6)
SSDΓ3n=600n2−1380n+791,(7)
HIΓ3n=1/15441n2−813n+3897,(8)
ISIΓ3n=1260n2−2892n+1650,(9)
AZIΓ3n=1008n2−1968n+960.Proof.
Let(31)MΓ3n;a,b=72n2−144n+72a2b4+42a2b7+60n−1a2b10+54n2−162n+126a2b18.
Now by applying a combination of the differential and integral operators provided in the derivation Table4 on M-polynomial, we get
(32)Da+Dbfa,b=672n2−144n+72a2b4+378a2b7+1260n−120a2b10+2054n2−162n+126a2b18.DaDbfa,b=872n2−144n+72a2b4+588a2b7+2060n−120a2b10+3654n2−162n+126a2b18.SaSbfa,b=9n2−18n+9a2b4+3a2b7+3n−6a2b10+123n2−9n+7a2b18.DaβDbβfa,b=8β72n2−144n+72a2b4+3×141+βa2b7+20β60n−120a2b10+36β54n2−162n+126a2b18.SaβSbβfa,b=81−β9n2−18n+9a2b4+3×141−βa2b7+201−β3n−6a2b10+61−2β9n2−27n+21a2b18.SbDa+SaDbfa,b336n2−72n+36a2b4+159a2b7+2612n−24a2b10+826n2−18n+14a2b18.Jfa,b=fa,a=72n2−2n+1a6+42a9+60n−2a12+183n2−9n+7a20.SaJfa,b=12n2−2n+1a6+143a9+5n−2a12+9103n2−9n+7a20SaJDaDbfa,b=96n2−2n+1a6+1963a9+100n−2a12+9554n2−162n+126a20.Sa3Q−2JDa3Db3fa,b=576n2−2n+1a4+336a7+480n−2a10+1443n2−9n+7a20.Now, by applying formulas of molecular descriptors derived from M-polynomial depicted in table??, we get:(1)
M1Γ3n=Da+Dbfa,ba=b=1=1512n2−3384n+1890,(2)
M2Γ3n=DbDafa,ba=b=1=2520n2−5784n+3300,(3)
mM2Γ3n=SaSbfa,ba=b=1=1/221n2−39n+19,(4)
RβΓ3n=DaβDbβfa,ba=b=1981+β+61+2βn2−36×81+β−201+β+9×61+2βn+33×81+β−2×201+β+141+β+7×61+2β,(5)
RRβΓ3n=SaβSbβfa,ba=b=1=981−β+61−2βn2−36×81−β−201−β+9×61−2βn+33×81−β−2×201−β+141−β+7×61−2β,(6)
SSDΓ3n=DaSb+SaDbf(a,b)a=b=1=600n2−1380n+791,(7)
HIΓ3n=2SaJfa,aa=1=1/15441n2−813n+3897,(8)
ISIΓ3n=SaJDaDbfa,ba=b=1=1260n2−2892n+1650,(9)
AZIΓ3n=Sa3Q−2JDa3Db3fa,ba=b=1=1008n2−1968n+960.


## 3.3. Results for Subdivided Hex-Derived NetworkSHDN1n.

In an earlier article [54], we studied and figured out numerous topological invariants of SHDN1n=Γ1n. A 4-dimensional subdivided hex-derived network of first kind Γ14 is shown in Figure 5. To perform a comparative analysis, the aforementioned results are presented here.Theorem 7 (see [54]).
LetΓ1ndenote the subdivided hex-derived network of first kind, and then(33)ABCΓ1n=1254n2−102n+48,GAΓ1n=612635n2+410524514−3786−40524n+41051896+22510−49014+31524,ABC4Γ1n=31905+37010+464n2+6267+1231266+641114−11464−91905−937010+977035n+6+26+125+127−18267−2431266−1241114+5462+317017+71905+32105−737010+344234−1877035,GA5Γ1n=33241091+6n2+9152−29161091+335+4811443+826611−54n+4837+1653+3241013−915+243011−635+244213−2410241−9611443+41709−1626611+24.Proposition 8 (see [54]).
SupposeΓ1ndenote subdivided hex-derived network of first type, and then, the first Zagreb, the second Zagreb, the modified Zagreb, the augmented Zagreb, the inverse Randic′, the general Randic′, the symmetric division degree, the inverse sum, and the harmonic indices are(1)
M1Γ1n=1833n2−73n+40,(2)
M2Γ1n=1281n2−185n+107,(3)
mM2Γ1n=1/29n2−15n+7,(4)
RβΓ1n=3×6β+11+22β+1n2+6×2β7β+1−2×3β+1−22β+1×3β+2η+310β+1+6β+1−2×14β+1+28×24β,(5)
RRβΓ1n=361−β+12×24−βn2+3141−β−2×61−β−36×24−βη+3101−β+61−β−2×141−β+28×24−β,(6)
SDDΓ1n=261n2−585n+326,(7)
HIΓ1n=4/105324n2−538n+239,(8)
ISIΓ1n=4/1052187n2−4279n+2042,(9)
AZIΓ1n=489n2−17n+8.Proposition 9.
LetΓ1n, Γ2n, andΓ3nbe the the hex-derived networks of type 1, type 2, and type 3, respectively. Then(34)χ−12Γ1n183575+514n2+235245−1265−13514n+235635+757+10514−490,χ−12Γ2n17632+1814n2−1892−307+5414−98n+1262+4214−196,χ−12Γ3n15606+275n2−1206−503+815n+606−1003+635+70.Figure 5
A 4-dimensional subdivided hex-derived network of a first kindΓ14.


## 4. Concluding Remarks: A Comparative Analysis of Subdivided Hex-Derived NetworksΓ1n, Γ2n and Γ3n

In the present article, we calculated various significant structure descriptors for subdivision of hex-derived networkSHDN2n and SHDN3n. These descriptors enable scientists to develop a better understanding of certain characteristics of chemical compounds and correlate different physicochemical as well as biological properties of these compounds. The subsequent figures play a key role in providing a comparative analysis between the subdivided hex-derived networks Γ1n, Γ2n, and Γ3n based upon various descriptors such as Zagreb indices (see Figures 6 and 7), sum connectivity index (see Figure 8), Randic index (see Figure 9), GA index (see Figure 10), HI (see Figure 11), AZI (see Figure 12), ABC index (see Figure 13), ISI (see Figure 14), SDD index (see Figure 15), GA5 index (see Figure 16), and ABC4 (see Figure 17).Figure 6
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of FZI.Figure 7
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of SZI.Figure 8
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of SCI.Figure 9
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of Randic′ index.Figure 10
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of GA index.Figure 11
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of HI.Figure 12
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of AZI.Figure 13
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of ABC index AZI.Figure 14
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of ISI index.Figure 15
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of SDD.Figure 16
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of GA5 index.Figure 17
Comparison ofΓ1n, Γ2n, and Γ3n using 2D graphs of ABC4 index.We know that the molecular descriptors like the first Zagreb index, the second Zagreb index, the Randic′ index, and the variations of Randic′ index such as sum-connectivity, atom-bond connectivity, and augmented Zagreb indices are used to measure the extent of connectivity in the molecular graph. Figure 6–17 admit the following order between the descriptors, TIΓ1n≤TIΓ2n≤TIΓ3n, where TI∈M1,M2,R−1/2,χ−1/2,ABC,AZI. It turns out that molecular graph Γ3n is more compact and dense as compared to Γ2n and Γ1n, whereas Γ2n is more compact and dense as compared to Γn. We anticipate the following ordering between the melting temperature (MT) of the compounds having a molecular structure similar to Γ1n, Γ2n, and Γ3n, MTΓ1n≤MTΓ2n≤MTΓ3n.Vukic˘evic′ and Gašperov [56] investigated the properties of 148 descriptors and observed that SDD and ISI are a significant predictors of the total surface area (TSA) of molecular graphs of certain chemical compounds. From Figure 15, we hypothesize the ordering in which total surface areas of Γ1n, Γ2n, and Γ3n could be arranged as follows: TSAΓ3n≤TSAΓ2n≤TSAΓ1n. Furthermore, the results presented in this paper might be helpful in the QSPR/QSAR analysis to predict further structural and physicochemical properties of molecular graphs under discussion.



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