# Study of Carbon NanoconesCNCkn via Connection Zagreb Indices

Authors: Muhammad Asif; Muhammad Hussain; Hamad Almohamedh; Khalid M Alhamed; Rana Alabdan; Abdulrazaq A. Almutairi; Sultan Almotairi
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

Topology of fullerenes, carbon nanotubes, and nanocones has considerable worth due to their effective applications in nanotechnology. These are emerging materials of practical application in gas storage devises, nanoelectronics devices, energy storage, biosensor, and chemical probes. The topological indices are graph invariant used to investigate the physical and chemical properties of the compounds such as boiling point, stability, and strain energy through associated chemical graph of the underlying compound. We computed recently modified Zagreb connection indices of nanoconesCNC4n,CNC5n, and CNC6n and generalized our findings up to a large class of CNCkn. Topological characterization of nanocones via these indices is mathematically novel and assists to enable its emerging use in nanotechnology. For computation and verification of results, we use Mathematica software.


---

## Body

## 1. Introduction

Carbon nanomaterials received considerable attention due to their effective physical applications in nanotechnology [1] as emerging materials of practical application. However, carbon nanocones CNCs received considerable attention after the discovery of free-standing structures or canonical topology as cap on one end of nanotubes CNTs [2, 3]. CNCs are considered as alternatives of CNTs due to the absence of potentially poisonous metal catalyst in synthesis and mass production at room temperature [4]. Generally, during the declamation of CNTs, strong acids are used in order to close out metal catalysts. In this process, deficiency is introduced with the hindrance of destructing the graphite structure. On the contrary, the applications and properties of CNCs are easy to approach. CNCs’ application as drug delivery capsules [5] and gas storage devices increased their significance. Throughout the years, this subject has been developing scientific obsession with planar, curved, and wrapped nanoscale structures, such as graphene, fullerenes, and nanotubes. It has a strong technological interest just because of their innovative structural, electronic, and mechanical properties. Curved carbon structures are used to investigate growth and nucleation. Especially, pentagon presence in CNCs plays a vital role [6]. The 60 declination defect is detected when pentagon inserted in the graphite sheet. This is the key of CNCs’ formation with pentagon as tip apex which leads us to the existence of nanotubes with tip topology. This type of defects in graphite networks is theoretically considered for study of electronic states [7]. CNCs have free-standing structures with sharp edges because these properties have applications in technology and electronics [8]. In Figure 1, the canonical form of CNCs is shown in Figure 1(a) and associated chemical graph is shown in Figure 1(b). We are interested in the characterization of CNCs using chemical graph theory.Figure 1
Carbon nanocones with associated chemical graph. (a) Canonical form of carbon nanocones. (b) Chemical Graph of carbon nanocones.
(a)(b)


### 1.1. Graph

LetG=V,E be a graph comprising set of vertices V and E⊆V2 as the set of edges. A graph is called the directed graph if the edges have some orientations. In a multigraph, two vertices can share more than one edge. A loop in a graph is an edge joining a vertex to itself. Graph is called simple if it is not directed or multigraph and has no loops.


### 1.2. Chemical Graph Theory

Graph theory is considered as a powerful tool in different areas of research such as in coding theory, database management system, circuit design, secret sharing schemes, and theoretical chemistry. Chemical graph theory is the combination of chemistry and graph theory. It develops a relationship between structure of organic substances and their physio-chemical properties through some useful graph invariants with the help of their associated molecular graph. The molecular graph is a simple graph on vertices which are representatives of atoms of corresponding chemical substance and edges placed against the bonds between atoms. Figure2 depicts chemical graphs of some hydrocarbons as benzene in Figure 2(a) and naphthalene in Figure 2(b). The theoretical study of underlying substance using molecular graphs through graph invariants has effective applications in quantitative structure properties relationship (QSPR) or quantitative structure activities relationship (QSAR) investigation [9].Figure 2
Some hydrocarbons with associated chemical graphs. (a) Benzene with chemical graph. (b) Naphthalene with chemical graph.
(a)(b)


## 1.1. Graph

LetG=V,E be a graph comprising set of vertices V and E⊆V2 as the set of edges. A graph is called the directed graph if the edges have some orientations. In a multigraph, two vertices can share more than one edge. A loop in a graph is an edge joining a vertex to itself. Graph is called simple if it is not directed or multigraph and has no loops.


## 1.2. Chemical Graph Theory

Graph theory is considered as a powerful tool in different areas of research such as in coding theory, database management system, circuit design, secret sharing schemes, and theoretical chemistry. Chemical graph theory is the combination of chemistry and graph theory. It develops a relationship between structure of organic substances and their physio-chemical properties through some useful graph invariants with the help of their associated molecular graph. The molecular graph is a simple graph on vertices which are representatives of atoms of corresponding chemical substance and edges placed against the bonds between atoms. Figure2 depicts chemical graphs of some hydrocarbons as benzene in Figure 2(a) and naphthalene in Figure 2(b). The theoretical study of underlying substance using molecular graphs through graph invariants has effective applications in quantitative structure properties relationship (QSPR) or quantitative structure activities relationship (QSAR) investigation [9].Figure 2
Some hydrocarbons with associated chemical graphs. (a) Benzene with chemical graph. (b) Naphthalene with chemical graph.
(a)(b)


## 2. Topological Indices

Topological indices among graph invariants have a special place and are used to estimate the physio-chemical properties of chemical compound. A topological index is considered as a functionf:G⟶R which maps each graph of chemical structures into a numerical value and have special place among other graph invariants due to its estimation applicability for physio-chemical properties of chemical compound. The idea of topological indices was first introduced by Wiener in 1947 during the work on paraffin’s boiling points [10]. In 1972, Zagreb indices were introduced by Ivan Gutman and Trinajstic [11]. Second Zagreb index was thought of by Hosoya et al. in 1975 [12]. The first and second Zagreb indices are defined as(1)M1Γ=∑uv∈EΓdegu+degv,M2Γ=∑uv∈EΓdegu×degv.Recently, modified versions of Zagreb indices were introduced and studied independently in [13–15]. These indices based on connection number are assigned to the vertices of graph. For more detail of topological indices, one can refer to [16–18].


### 2.1. Connection Number

LetG be a graph. The connection number associated to the vertex u∈VG is the number of distinct vertices at distance two from vertex u. It is denoted by τu:(2)τu=v∈VG:du,v=2.


### 2.2. Connection Zagreb Indices

The firstZC1G and second ZC2G connection Zagreb indices are defined as(3)ZC1G=∑u∈VGτu2,ZC2G=∑uv∈EGτu×τv,where ZC1∗ is another recently introduced graph invariant over connection number [19]:(4)ZC1∗G=∑u∈VGdeguτu=∑uv∈EGτu+τv,where degu stands for degree of u∈G and τu is the connection number assigned to u. Ali et al. [19] proved that, for triangle and rectangle free graphs, connection number τu assigned to a vertex u is τu=∑uv∈EGdegv−degu.


### 2.3. Applications of Connection Zagreb Indices in Chemistry

Applicability ofZC1G,ZC2G, and ZC1∗G is observed by its good correlation with entropy of octane isomers [19]. Ali et al. [20] concluded that ZC1∗ has correlation coefficient approximately 0.949 and 0.892 for acentric factor and entropy, respectively, and ZC1 has a good correlation with enthalpy. Javaid et al. studied T-sum graphs and graphs constructed through graph operation using connection indices [21–25]. This contemplation recommends chemical applicability of these indices as useful descriptor in QSAR and QSPR investigation.


## 2.1. Connection Number

LetG be a graph. The connection number associated to the vertex u∈VG is the number of distinct vertices at distance two from vertex u. It is denoted by τu:(2)τu=v∈VG:du,v=2.


## 2.2. Connection Zagreb Indices

The firstZC1G and second ZC2G connection Zagreb indices are defined as(3)ZC1G=∑u∈VGτu2,ZC2G=∑uv∈EGτu×τv,where ZC1∗ is another recently introduced graph invariant over connection number [19]:(4)ZC1∗G=∑u∈VGdeguτu=∑uv∈EGτu+τv,where degu stands for degree of u∈G and τu is the connection number assigned to u. Ali et al. [19] proved that, for triangle and rectangle free graphs, connection number τu assigned to a vertex u is τu=∑uv∈EGdegv−degu.


## 2.3. Applications of Connection Zagreb Indices in Chemistry

Applicability ofZC1G,ZC2G, and ZC1∗G is observed by its good correlation with entropy of octane isomers [19]. Ali et al. [20] concluded that ZC1∗ has correlation coefficient approximately 0.949 and 0.892 for acentric factor and entropy, respectively, and ZC1 has a good correlation with enthalpy. Javaid et al. studied T-sum graphs and graphs constructed through graph operation using connection indices [21–25]. This contemplation recommends chemical applicability of these indices as useful descriptor in QSAR and QSPR investigation.


## 3. Materials and Methods

In this work, we use graph theoretic techniques adopted in [21–25] to study the topology of the corresponding molecular graph of underlying compound for their insight investigation. We used investigative procedure, vertex segment strategy, edge segment procedure, and degree tallying strategy along with combinatorial enlisting techniques, number theoretic logics, and edges and vertices partition according to the values associated to the vertices for desired computation.Throughout this work, we use standard graph theoretic notations, graphG=V,E, set of vertices VG and the set of edges EG of a graph G with V order and E size of G, degvi as degree for vertex vi, and the number of edges incident to vi. We draw graph of CNCkn using Mathematica software for k=4,5,6 and compute degree sequence for different values of n. Through observations, we listed possibilities for connection numbers assigned to the vertices and determined vertex petition as well as edge partitions with respect to connection number assigned to the vertices of these graphs.


## 4. Results and Discussion

In this work, we compute connection Zagreb indicesZC1G,ZC2G, and ZC1∗G of CNCs which are helpful in their topological investigation and generalize our findings up to a large class of CNCkn. CNCkn are classified on the basis of k, where k is the number of carbon atoms present in the core of nanocones and n is the number of hexagon layers around the core [26].


### 4.1. Results forCNC4n

In [27], Ghorbani and Jalali compute vertex PI index, Szeged index, and Omega Polynomials, and Hayat and Imran [28] computed ABC4 and GA5 of CNC4n. In Theorem 1, we computed connection Zagreb indices of CNC4n. CNC4n for n=3 is shown in Figure 3.Figure 3
CNC43.Theorem 1.
TheZC1,ZC2, and ZC1∗ of CNC4n are(5)ZC1CNC4n=144n2+128n+8,ZC2CNC4n=216n2+152n+4,ZC1∗CNC4n=72n2+80n+16.Proof.
LetG=CNC4n be the graph of under consideration CNCs. The total number of vertices of CNC4n are 4n+12 and 23n2+5n+2 edges. We made partition of vertex set with respect to the connection number τu as VτuG. For this purpose, we draw graphs of CNC4n for n=1,2,3 using Mathematica and compute degree sequence as 2,3. Degree sequence 2,3 of CNC4n implies 3,4, and 6 are the only possibilities of the connection numbers associated to the vertices of G and 3,3,3,4,4,4,4,6, and 6,6 are the possibilities for the edges. For partition of edges having the same end vertices connection numbers, we use edge and vertex segment strategy along with edges and vertex listing technique. The following tabular calculation enables us for this partition through induction.
The numerical results of Table1 depict V3G=8,V4G=8n−4, and V6G=4n2. By using this partition, we computed ZC1CNC4n:(6)ZC1G=∑u∈VGτu2.
LetNτ=τu:u∈VG, and from vertex partition VτuG, we obtain(7)ZC1G=∑τu∈Nτ∑u∈VτuGτu=∑u∈V3Gτu+∑u∈V4Gτu+∑u∈V6Gτu,ZC1G=832+8n−442+4n262,ZC1CNC4n=144n2+168n−4+72.
In Table2, we list our observation for the number of edges with similar end vertex connection number and generalize our findings for arbitrary value of n.
The generalization based on Table2 provides edge partitions as Eτu,τvG:E3,3G=4,E3,4G=8,E4,4G=8n−1,E4,6G=4n, and E6,6G=2n3n−1. Using this partition, we computed ZC2CNCkn defined as(8)ZC2G=∑uv∈EGτuτv.
LetMτ=τu,τv:uv∈G:(9)ZC2G=∑τu,τv∈Mτ∑uv∈Eτu,τvGτuτv=∑uv∈E3,3Gτuτv+∑uv∈E3,4Gτuτv+∑uv∈E4,4Gτuτv+∑uv∈E4,6Gτuτv+∑uv∈E6,6Gτuτv,ZC2G=43×3+83×4+8n−84×4+4n4×6+2n3n−16×6,ZC2G=216n2+152n+4.
Now, we computeZC1∗G which is defined as(10)ZC1∗G=∑u∈VGτudegu=∑uv∈Eu,vGτu+τv.
LetOτ=τu,τv:uv∈EG and(11)ZC1∗G=∑τu,τv∈Oτ∑uv∈Eτu,τvGτu+τv=∑uv∈E3,3Gτu+τv+∑uv∈E3,4Gτu+τv+∑uv∈E4,4Gτu+τv+∑uv∈E4,6Gτu+τv+∑uv∈E6,6Gτu+τv,s,ZC1∗G=43+3+83+4+8n−84+4+4n4+6+2n3n−16+6,ZC1∗G=72n2+80n+16.Table 1
Vertex partition ofCNC4n with respect to connection number τ.
n12345678910…pτu=38888888888…8τu=44122028384452606876…8p−4τu=64163664100144196256324400…4p2Table 2
Edge partition ofCNC4n with respect to connection number τ.
n/Eτu,τv12345678910…n=pE3,34444444444…4E3,48888888888…8E4,4081624324048566472…8(p-1)E4,6481216202428323640…4pE6,64204888140204280368468580…2p3p−1


### 4.2. Results forCNC5n

A. R. Ashrafi in [29] computed winner index of CNC5n. In Theorem 2, we computed connection Zagreb indices of CNC5n whose chemical graph for n=3 is shown in Figure 4.Figure 4
Chemical graph ofCNC5n for n=3.Theorem 2.
TheZC1,ZC2, and ZC1∗ of CNC5n are equal to(12)ZC1CNC5n=180n2+160n+10,ZC2CNC5n=270n2+190n+5,ZC1∗CNC5n=90n2+100n+20.Proof.
The total number of vertices ofG=CNC5n are 5n+12 and edges 5/23n2+5n+2. Like CNC4n, we partitioned the vertex set as VτuG and the edge set as Eτu,τvG by using edge and vertex segment strategy along with edges and vertex listing technique for desired partition.
The computations in Table3 depict V3G=10,V4G=10n−5, and V6G=5n2. By using this partition, we compute ZC1CNC5n=ZC1G as(13)ZC1G=∑u∈VGτu2.
LetNτ=τu:u∈VG; then, by using vertex set partition Vτu, we obtain(14)ZC1G=∑τu∈Nτ∑u∈VτuGτu=∑u∈V3Gτu+∑u∈V4Gτu+∑u∈V6Gτu,ZC1G=1032+10n−542+5n262,ZC1CNC5n=180n2+160n+10.
In Table4, we list our observation for the number of edges with similar end vertex connection number and generalize our findings for n.
The generalization based on Table4 provides edge partitions as Eτu,τvG:E3,3G=5,E3,4G=10,E4,4G=10n−1,E4,6G=5n, and E6,6G=5/2n3n−1. Using this partition, we computed ZC2CNC5n and ZC1∗CNC5n:(15)ZC2G=∑uv∈EGτuτv.
LetMτ=τu,τv:uv∈G; then, by using vertex edge partition Eτu,τvG, we obtain(16)ZC2G=∑τu,τv∈Mτ∑uv∈Eτu,τvGτuτv=∑uv∈E3,3Gτuτv+∑uv∈E3,4Gτuτv+∑uv∈E4,4Gτuτv+∑uv∈E4,6Gτuτv+∑uv∈E6,6Gτuτv=53×3+103×4+10n−104×4+5n4×6+5/2n3n−16×6=270n2+190n+5.
ForZC1∗G,(17)ZC1∗G=∑u∈VGτudegu=∑uv∈Eu,vGτu+τv.
LetOτ=τu,τv:uv∈EG:(18)ZC1∗G=∑τu,τv∈Oτ∑uv∈Eτu,τvGτu+τv=∑uv∈E3,3Gτu+τv+∑uv∈E3,4Gτu+τv+∑uv∈E4,4Gτu+τv+∑uv∈E4,6Gτu+τv+∑uv∈E6,6Gτu+τv=53+3+103+4+10n−104+4+5n4+6+52n3n−16+6=90n2+100n+20.Table 3
Vertex partition ofCNC5n with respect to connection number τ.
n12345678910…pτ=310101010101010101010…10τ=45152535455565758595…10p−5τ=65204580125180245320405500…5p2Table 4
Edge partition ofCNC5n with respect to connection number τ.
n/Eτu,τv12345678910…n=pE3,35555555555…5E3,410101010101010101010…10E4,40102030405060708090…10(p-1)E4,65101520253035404550…5pE6,652560110175255350460585725…5p/23p−1


### 4.3. Results forCNC6n

In Theorem3, we computed connection Zagreb indices of CNCkn, for k=6. CNC6n for n=4 is shown in Figure 5.Figure 5
Graphical representation ofCNC6n.Theorem 3.
TheZC1,ZC2, and ZC1∗ of CNC5n are equal to(19)ZC1CNC6n=180n2+160n+10,ZC2CNC6n=270n2+190n+5,ZC1∗CNC6n=108n2+100n+20.Proof.
The total number of vertices ofG=CNC6n are 6n+12 and 33n2+5n+2 edges. Similar technique is used as in CNC4n and CNC5n computation. We partitioned the vertex set as VτuG and the edge set as Eτu,τvG.
The computations in Table5 give vertex partition V3G=10,V4G=10n−5, and V6G=5n2. By using this partition, we compute ZC1CNC6n=ZC1G:(20)ZC1G=∑u∈VGτu2.
LetNτ=τu:u∈VG:(21)ZC1G=∑τu∈Nτ∑u∈VτuGτu=∑u∈V3Gτu+∑u∈V4Gτu+∑u∈V6Gτu,ZC1G=1232+12n−642+6n262,ZC1CNC6n=216n2+192n+12.
In Table6, we list our observation for the number of edges with similar end vertex connection number of CNC6n and generalize for arbitrary value of n.
The generalization based on Table6 provides edge partitions as Eτu,τvG,E3,3G=6,E3,4G=12,E4,4G=12n−1,E4,6G=6n, and E6,6G=3n3n−1. Using this partition, we computed ZC2CNC6n and ZC1∗CNC6n:(22)ZC2G=∑uv∈EGτuτv,
LetMτ=τu,τv:uv∈G,(23)ZC2G=∑τu,τv∈Mτ∑uv∈Eτu,τvGτuτv=∑uv∈E3,3Gτuτv+∑uv∈E3,4Gτuτv+∑uv∈E4,4Gτuτv+∑uv∈E4,6Gτuτv+∑uv∈E6,6Gτuτv=63×3+123×4+12n−124×4+6n4×6+3n3n−16×6,ZC2CNC6n=324n2+228n+6.
Now, forZC1∗G,(24)ZC1∗G=∑u∈VGτudegu=∑uv∈Eu,vGτu+τv.
LetOτ=τu,τv:uv∈EG:(25)ZC∗G=∑τu,τv∈Oτ∑uv∈Eτu,τvGτu+τv=∑uv∈E3,3Gτu+τv+∑uv∈E3,4Gτu+τv+∑uv∈E4,4Gτu+τv+∑uv∈E4,6Gτu+τv+∑uv∈E6,6Gτu+τv=63+3+123+4+12n−124+4+6n4+6+3n3n−16+6,ZC1∗CNC6n=108n2+192n+24.Table 5
Vertex partition ofCNC6n with respect to connection number τ.
n12345678910…pτ=312121212121212121212…12τ=4618304254667890102114…12p−6τ=65245496150216294384486600…6p2Table 6
Edge partition ofCNC6n with respect to connection number τ.
n/Eτu,τv12345678910…n=pE3,36666666666…6E3,412121212121212121212…12E4,401224364860728496108…12(p-1)E4,66121824303642485460…6pE6,663072132210306420552702870…3p3p−1


### 4.4. Comparison of Results

In Figure6, Figure 6(a) interprets behavior of ZC1G, Figure 6(b) of ZC2G, and Figure 6(c) of ZC1∗G for k=4,5,6.Figure 6
Comparison of results computed in Theorems1–3. (a) Comparison of results regarding ZC1. (b) Comparison of results regarding ZC2. (c) Comparison of results regarding ZC1∗.
(a)(b)(c)


### 4.5. Results forCNCkn

Graph shown in Figure7 is the associated chemical graph of CNCkn for arbitrary value of k. It consists of a k-gon as the central core cycle with n layers of hexagons around the core.Figure 7
The chemical graph of generalized carbon nanoconeCNCkn.Theorem 4.
TheZC1,ZC2, and ZC1∗ of CNCkn are(26)ZC1CNCkn=36kn2+32kn+2k,ZC2CNCkn=54kn2+38kn+k,ZC1∗CNCkn=18kn2+20kn+4k.Proof.
The chemical graph ofCNCkn has cycle of length k corresponding to core of CNCkn, where n represent the number of layers of hexagons around the core. Total vertices of CNCkn is kn+12 and edges is k/23n2+5n+2 [30]. For computation of ZC1 for generalized nanocone CNCkn, we used their edges and vertex partition. From the construction of CNCkn, it is clear that the connection number associated to the vertices is 6 except the k2n+1 vertices of outer layer of hexagons. So, the number of vertices with connection number 6 is V6CNCkn=kn+12−k2n+1=kn2=kn2.
In the outer layer of hexagons, two vertices at each corner have connection numberτu=3 and all other have τu=4. So, V3CNCkn=2k and V4CNCkn=k2n+1−2k=k2n−1. Similarly, all the edges other than edges of the outer layer of hexagons have 6,6 as end vertex connection number, i.e., E6,6CNCkn=k/23n2+5n+2−3kn+k=k/23n2−1. The edges in the outer layer of hexagons are partitioned as E3,3CNCkn=k,E3,4CNCkn=2k,E4,4CNCkn=k2n−2, and E4,6CNCkn=kn. We use this vertex and edge partition and technique as used in Theorems 1–3 for proof. Let G=CNCKn and Nτ=τu:u∈VG:(27)ZC1G=∑u∈VGτu2,ZC1G=∑τu∈Nτ∑u∈VτuGτu=∑u∈V3Gτu+∑u∈V4Gτu+∑u∈V6Gτu,ZC1G=2k32+k2n−142+kn262,ZC1CNCkn=36kn2+32kn+2k,(28)ZC2G=∑uv∈EGτuτv,ZC2G=∑τu,τv∈Mτ∑uv∈Eτu,τvGτuτv=∑uv∈E3,3Gτuτv+∑uv∈E3,4Gτuτv+∑uv∈E4,4Gτuτv+∑uv∈E4,6Gτuτv+∑uv∈E6,6Gτuτv,ZC2G=k3×3+2k3×4+2kn−2k4×4+kn4×6+k23n2−16×6,ZC1CNCkn=54kn2+38kn+k,(29)ZC1∗G=∑u∈VGτudegu=∑uv∈Eu,vGτu+τv,ZC1∗G=∑τu,τv∈Oτ∑uv∈Eτu,τvGτu+τv=∑uv∈E3,3Gτu+τv+∑uv∈E3,4Gτu+τv+∑uv∈E4,4Gτu+τv+∑uv∈E4,6Gτu+τv+∑uv∈E6,6Gτu+τv,ZC1∗G=k3+3+2k3+4+2kn−2k4+4+kn4+6+k23n2−16+6,ZC1∗CNCkn=18kn2+20kn+4k.
Hence, the proof is completed.
Theorem4 shows that connection Zagreb indices of CNCkn increased by a factor k (number of carbon atoms in the core). In Figure 8, Figures 8(a) and 8(b) interpret computed results of ZC1CNCkn and ZC2CNCkn, respectively, for any value of k and 9 of ZC1∗CNCkn.Figure 8
The graphical representation of connection Zagreb indices for generalized values ofn and k. (a) The graphical representation of ZC1(CNCk(n)). (b) The graphical representation of ZC2(CNCk(n)).
(a)(b)Figure 9
The graphical representation ofZC1∗CNCkn for generalized values k.


## 4.1. Results forCNC4n

In [27], Ghorbani and Jalali compute vertex PI index, Szeged index, and Omega Polynomials, and Hayat and Imran [28] computed ABC4 and GA5 of CNC4n. In Theorem 1, we computed connection Zagreb indices of CNC4n. CNC4n for n=3 is shown in Figure 3.Figure 3
CNC43.Theorem 1.
TheZC1,ZC2, and ZC1∗ of CNC4n are(5)ZC1CNC4n=144n2+128n+8,ZC2CNC4n=216n2+152n+4,ZC1∗CNC4n=72n2+80n+16.Proof.
LetG=CNC4n be the graph of under consideration CNCs. The total number of vertices of CNC4n are 4n+12 and 23n2+5n+2 edges. We made partition of vertex set with respect to the connection number τu as VτuG. For this purpose, we draw graphs of CNC4n for n=1,2,3 using Mathematica and compute degree sequence as 2,3. Degree sequence 2,3 of CNC4n implies 3,4, and 6 are the only possibilities of the connection numbers associated to the vertices of G and 3,3,3,4,4,4,4,6, and 6,6 are the possibilities for the edges. For partition of edges having the same end vertices connection numbers, we use edge and vertex segment strategy along with edges and vertex listing technique. The following tabular calculation enables us for this partition through induction.
The numerical results of Table1 depict V3G=8,V4G=8n−4, and V6G=4n2. By using this partition, we computed ZC1CNC4n:(6)ZC1G=∑u∈VGτu2.
LetNτ=τu:u∈VG, and from vertex partition VτuG, we obtain(7)ZC1G=∑τu∈Nτ∑u∈VτuGτu=∑u∈V3Gτu+∑u∈V4Gτu+∑u∈V6Gτu,ZC1G=832+8n−442+4n262,ZC1CNC4n=144n2+168n−4+72.
In Table2, we list our observation for the number of edges with similar end vertex connection number and generalize our findings for arbitrary value of n.
The generalization based on Table2 provides edge partitions as Eτu,τvG:E3,3G=4,E3,4G=8,E4,4G=8n−1,E4,6G=4n, and E6,6G=2n3n−1. Using this partition, we computed ZC2CNCkn defined as(8)ZC2G=∑uv∈EGτuτv.
LetMτ=τu,τv:uv∈G:(9)ZC2G=∑τu,τv∈Mτ∑uv∈Eτu,τvGτuτv=∑uv∈E3,3Gτuτv+∑uv∈E3,4Gτuτv+∑uv∈E4,4Gτuτv+∑uv∈E4,6Gτuτv+∑uv∈E6,6Gτuτv,ZC2G=43×3+83×4+8n−84×4+4n4×6+2n3n−16×6,ZC2G=216n2+152n+4.
Now, we computeZC1∗G which is defined as(10)ZC1∗G=∑u∈VGτudegu=∑uv∈Eu,vGτu+τv.
LetOτ=τu,τv:uv∈EG and(11)ZC1∗G=∑τu,τv∈Oτ∑uv∈Eτu,τvGτu+τv=∑uv∈E3,3Gτu+τv+∑uv∈E3,4Gτu+τv+∑uv∈E4,4Gτu+τv+∑uv∈E4,6Gτu+τv+∑uv∈E6,6Gτu+τv,s,ZC1∗G=43+3+83+4+8n−84+4+4n4+6+2n3n−16+6,ZC1∗G=72n2+80n+16.Table 1
Vertex partition ofCNC4n with respect to connection number τ.
n12345678910…pτu=38888888888…8τu=44122028384452606876…8p−4τu=64163664100144196256324400…4p2Table 2
Edge partition ofCNC4n with respect to connection number τ.
n/Eτu,τv12345678910…n=pE3,34444444444…4E3,48888888888…8E4,4081624324048566472…8(p-1)E4,6481216202428323640…4pE6,64204888140204280368468580…2p3p−1


## 4.2. Results forCNC5n

A. R. Ashrafi in [29] computed winner index of CNC5n. In Theorem 2, we computed connection Zagreb indices of CNC5n whose chemical graph for n=3 is shown in Figure 4.Figure 4
Chemical graph ofCNC5n for n=3.Theorem 2.
TheZC1,ZC2, and ZC1∗ of CNC5n are equal to(12)ZC1CNC5n=180n2+160n+10,ZC2CNC5n=270n2+190n+5,ZC1∗CNC5n=90n2+100n+20.Proof.
The total number of vertices ofG=CNC5n are 5n+12 and edges 5/23n2+5n+2. Like CNC4n, we partitioned the vertex set as VτuG and the edge set as Eτu,τvG by using edge and vertex segment strategy along with edges and vertex listing technique for desired partition.
The computations in Table3 depict V3G=10,V4G=10n−5, and V6G=5n2. By using this partition, we compute ZC1CNC5n=ZC1G as(13)ZC1G=∑u∈VGτu2.
LetNτ=τu:u∈VG; then, by using vertex set partition Vτu, we obtain(14)ZC1G=∑τu∈Nτ∑u∈VτuGτu=∑u∈V3Gτu+∑u∈V4Gτu+∑u∈V6Gτu,ZC1G=1032+10n−542+5n262,ZC1CNC5n=180n2+160n+10.
In Table4, we list our observation for the number of edges with similar end vertex connection number and generalize our findings for n.
The generalization based on Table4 provides edge partitions as Eτu,τvG:E3,3G=5,E3,4G=10,E4,4G=10n−1,E4,6G=5n, and E6,6G=5/2n3n−1. Using this partition, we computed ZC2CNC5n and ZC1∗CNC5n:(15)ZC2G=∑uv∈EGτuτv.
LetMτ=τu,τv:uv∈G; then, by using vertex edge partition Eτu,τvG, we obtain(16)ZC2G=∑τu,τv∈Mτ∑uv∈Eτu,τvGτuτv=∑uv∈E3,3Gτuτv+∑uv∈E3,4Gτuτv+∑uv∈E4,4Gτuτv+∑uv∈E4,6Gτuτv+∑uv∈E6,6Gτuτv=53×3+103×4+10n−104×4+5n4×6+5/2n3n−16×6=270n2+190n+5.
ForZC1∗G,(17)ZC1∗G=∑u∈VGτudegu=∑uv∈Eu,vGτu+τv.
LetOτ=τu,τv:uv∈EG:(18)ZC1∗G=∑τu,τv∈Oτ∑uv∈Eτu,τvGτu+τv=∑uv∈E3,3Gτu+τv+∑uv∈E3,4Gτu+τv+∑uv∈E4,4Gτu+τv+∑uv∈E4,6Gτu+τv+∑uv∈E6,6Gτu+τv=53+3+103+4+10n−104+4+5n4+6+52n3n−16+6=90n2+100n+20.Table 3
Vertex partition ofCNC5n with respect to connection number τ.
n12345678910…pτ=310101010101010101010…10τ=45152535455565758595…10p−5τ=65204580125180245320405500…5p2Table 4
Edge partition ofCNC5n with respect to connection number τ.
n/Eτu,τv12345678910…n=pE3,35555555555…5E3,410101010101010101010…10E4,40102030405060708090…10(p-1)E4,65101520253035404550…5pE6,652560110175255350460585725…5p/23p−1


## 4.3. Results forCNC6n

In Theorem3, we computed connection Zagreb indices of CNCkn, for k=6. CNC6n for n=4 is shown in Figure 5.Figure 5
Graphical representation ofCNC6n.Theorem 3.
TheZC1,ZC2, and ZC1∗ of CNC5n are equal to(19)ZC1CNC6n=180n2+160n+10,ZC2CNC6n=270n2+190n+5,ZC1∗CNC6n=108n2+100n+20.Proof.
The total number of vertices ofG=CNC6n are 6n+12 and 33n2+5n+2 edges. Similar technique is used as in CNC4n and CNC5n computation. We partitioned the vertex set as VτuG and the edge set as Eτu,τvG.
The computations in Table5 give vertex partition V3G=10,V4G=10n−5, and V6G=5n2. By using this partition, we compute ZC1CNC6n=ZC1G:(20)ZC1G=∑u∈VGτu2.
LetNτ=τu:u∈VG:(21)ZC1G=∑τu∈Nτ∑u∈VτuGτu=∑u∈V3Gτu+∑u∈V4Gτu+∑u∈V6Gτu,ZC1G=1232+12n−642+6n262,ZC1CNC6n=216n2+192n+12.
In Table6, we list our observation for the number of edges with similar end vertex connection number of CNC6n and generalize for arbitrary value of n.
The generalization based on Table6 provides edge partitions as Eτu,τvG,E3,3G=6,E3,4G=12,E4,4G=12n−1,E4,6G=6n, and E6,6G=3n3n−1. Using this partition, we computed ZC2CNC6n and ZC1∗CNC6n:(22)ZC2G=∑uv∈EGτuτv,
LetMτ=τu,τv:uv∈G,(23)ZC2G=∑τu,τv∈Mτ∑uv∈Eτu,τvGτuτv=∑uv∈E3,3Gτuτv+∑uv∈E3,4Gτuτv+∑uv∈E4,4Gτuτv+∑uv∈E4,6Gτuτv+∑uv∈E6,6Gτuτv=63×3+123×4+12n−124×4+6n4×6+3n3n−16×6,ZC2CNC6n=324n2+228n+6.
Now, forZC1∗G,(24)ZC1∗G=∑u∈VGτudegu=∑uv∈Eu,vGτu+τv.
LetOτ=τu,τv:uv∈EG:(25)ZC∗G=∑τu,τv∈Oτ∑uv∈Eτu,τvGτu+τv=∑uv∈E3,3Gτu+τv+∑uv∈E3,4Gτu+τv+∑uv∈E4,4Gτu+τv+∑uv∈E4,6Gτu+τv+∑uv∈E6,6Gτu+τv=63+3+123+4+12n−124+4+6n4+6+3n3n−16+6,ZC1∗CNC6n=108n2+192n+24.Table 5
Vertex partition ofCNC6n with respect to connection number τ.
n12345678910…pτ=312121212121212121212…12τ=4618304254667890102114…12p−6τ=65245496150216294384486600…6p2Table 6
Edge partition ofCNC6n with respect to connection number τ.
n/Eτu,τv12345678910…n=pE3,36666666666…6E3,412121212121212121212…12E4,401224364860728496108…12(p-1)E4,66121824303642485460…6pE6,663072132210306420552702870…3p3p−1


## 4.4. Comparison of Results

In Figure6, Figure 6(a) interprets behavior of ZC1G, Figure 6(b) of ZC2G, and Figure 6(c) of ZC1∗G for k=4,5,6.Figure 6
Comparison of results computed in Theorems1–3. (a) Comparison of results regarding ZC1. (b) Comparison of results regarding ZC2. (c) Comparison of results regarding ZC1∗.
(a)(b)(c)


## 4.5. Results forCNCkn

Graph shown in Figure7 is the associated chemical graph of CNCkn for arbitrary value of k. It consists of a k-gon as the central core cycle with n layers of hexagons around the core.Figure 7
The chemical graph of generalized carbon nanoconeCNCkn.Theorem 4.
TheZC1,ZC2, and ZC1∗ of CNCkn are(26)ZC1CNCkn=36kn2+32kn+2k,ZC2CNCkn=54kn2+38kn+k,ZC1∗CNCkn=18kn2+20kn+4k.Proof.
The chemical graph ofCNCkn has cycle of length k corresponding to core of CNCkn, where n represent the number of layers of hexagons around the core. Total vertices of CNCkn is kn+12 and edges is k/23n2+5n+2 [30]. For computation of ZC1 for generalized nanocone CNCkn, we used their edges and vertex partition. From the construction of CNCkn, it is clear that the connection number associated to the vertices is 6 except the k2n+1 vertices of outer layer of hexagons. So, the number of vertices with connection number 6 is V6CNCkn=kn+12−k2n+1=kn2=kn2.
In the outer layer of hexagons, two vertices at each corner have connection numberτu=3 and all other have τu=4. So, V3CNCkn=2k and V4CNCkn=k2n+1−2k=k2n−1. Similarly, all the edges other than edges of the outer layer of hexagons have 6,6 as end vertex connection number, i.e., E6,6CNCkn=k/23n2+5n+2−3kn+k=k/23n2−1. The edges in the outer layer of hexagons are partitioned as E3,3CNCkn=k,E3,4CNCkn=2k,E4,4CNCkn=k2n−2, and E4,6CNCkn=kn. We use this vertex and edge partition and technique as used in Theorems 1–3 for proof. Let G=CNCKn and Nτ=τu:u∈VG:(27)ZC1G=∑u∈VGτu2,ZC1G=∑τu∈Nτ∑u∈VτuGτu=∑u∈V3Gτu+∑u∈V4Gτu+∑u∈V6Gτu,ZC1G=2k32+k2n−142+kn262,ZC1CNCkn=36kn2+32kn+2k,(28)ZC2G=∑uv∈EGτuτv,ZC2G=∑τu,τv∈Mτ∑uv∈Eτu,τvGτuτv=∑uv∈E3,3Gτuτv+∑uv∈E3,4Gτuτv+∑uv∈E4,4Gτuτv+∑uv∈E4,6Gτuτv+∑uv∈E6,6Gτuτv,ZC2G=k3×3+2k3×4+2kn−2k4×4+kn4×6+k23n2−16×6,ZC1CNCkn=54kn2+38kn+k,(29)ZC1∗G=∑u∈VGτudegu=∑uv∈Eu,vGτu+τv,ZC1∗G=∑τu,τv∈Oτ∑uv∈Eτu,τvGτu+τv=∑uv∈E3,3Gτu+τv+∑uv∈E3,4Gτu+τv+∑uv∈E4,4Gτu+τv+∑uv∈E4,6Gτu+τv+∑uv∈E6,6Gτu+τv,ZC1∗G=k3+3+2k3+4+2kn−2k4+4+kn4+6+k23n2−16+6,ZC1∗CNCkn=18kn2+20kn+4k.
Hence, the proof is completed.
Theorem4 shows that connection Zagreb indices of CNCkn increased by a factor k (number of carbon atoms in the core). In Figure 8, Figures 8(a) and 8(b) interpret computed results of ZC1CNCkn and ZC2CNCkn, respectively, for any value of k and 9 of ZC1∗CNCkn.Figure 8
The graphical representation of connection Zagreb indices for generalized values ofn and k. (a) The graphical representation of ZC1(CNCk(n)). (b) The graphical representation of ZC2(CNCk(n)).
(a)(b)Figure 9
The graphical representation ofZC1∗CNCkn for generalized values k.


## 5. Conclusion

The International Academy of Theoretical Chemistry explored that any topological index is acceptable to estimate chemical properties of compounds if it has a sound correlation with actual physio-chemical property of octane isomers. Customarily, octane isomers are adopted for such investigations due to the huge number of structural isomers of octane [31] for the acceptable statistical conclusion. Ali et al. [32] studied the correlation efficiency of ZC1∗ for the physio-chemical properties of octane isomers: entropy, boiling point, density, enthalpies, acentric factor, etc. They concluded that ZC1∗ has a correlation coefficient approximately 0.949 and 0.892 for acentric factor and entropy, respectively. Correlation of ZC1 with physical properties was studied by Jakkannavar and Basavanagoud [33] and found good with enthalpy. Possibility of large-scale production of CNCs [34] and predictive ability of topological indices [35] encourages theoretical study of CNCkn. We computed connection Zagreb indices ZC1G,ZC2G, and ZC1∗G of CNCkn for k=4,5,6 in Theorems 1–3 and generalized our finding up to large value of K in 4.4. These results facilitate topological characterization of CNCkn which is helpful in finding new applications in the emerging field of nanotechnology.



---

*Source: 5539904-2021-05-12.xml*