#### 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 nanocones , and and generalized our findings up to a large class of . 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.

#### 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 received considerable attention after the discovery of free-standing structures or canonical topology as cap on one end of nanotubes [2, 3]. are considered as alternatives of due to the absence of potentially poisonous metal catalyst in synthesis and mass production at room temperature [4]. Generally, during the declamation of , 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 are easy to approach. â€™ 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 plays a vital role [6]. The declination defect is detected when pentagon inserted in the graphite sheet. This is the key of â€™ 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]. have free-standing structures with sharp edges because these properties have applications in technology and electronics [8]. In Figure 1, the canonical form of is shown in Figure 1(a) and associated chemical graph is shown in Figure 1(b). We are interested in the characterization of using chemical graph theory.

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##### 1.1. Graph

Let be a graph comprising set of vertices and 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. Figure 2 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].

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#### 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 function 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 paraffinâ€™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

Recently, modified versions of Zagreb indices were introduced and studied independently in [13â€“15]. These indices based on connection number are assigned to the vertices of graph. For more detail of topological indices, one can refer to [16â€“18].

##### 2.1. Connection Number

Let be a graph. The connection number associated to the vertex is the number of distinct vertices at distance two from vertex . It is denoted by :

##### 2.2. Connection Zagreb Indices

The first and second connection Zagreb indices are defined aswhere is another recently introduced graph invariant over connection number [19]:where stands for degree of and is the connection number assigned to . Ali et al. [19] proved that, for triangle and rectangle free graphs, connection number assigned to a vertex is .

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

Applicability of , and is observed by its good correlation with entropy of octane isomers [19]. Ali et al. [20] concluded that has correlation coefficient approximately and for acentric factor and entropy, respectively, and has a good correlation with enthalpy. Javaid et al. studied T-sum graphs and graphs constructed through graph operation using connection indices [21â€“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 [21â€“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, graph , set of vertices and the set of edges of a graph with order and size of , as degree for vertex , and the number of edges incident to . We draw graph of using Mathematica software for and compute degree sequence for different values of . 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 indices , and of CNCs which are helpful in their topological investigation and generalize our findings up to a large class of . are classified on the basis of , where 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 for

In [27], Ghorbani and Jalali compute vertex PI index, Szeged index, and Omega Polynomials, and Hayat and Imran [28] computed and of . In Theorem 1, we computed connection Zagreb indices of . for is shown in Figure 3.

Theorem 1. *The , and of are*

*Proof. *Let be the graph of under consideration . The total number of vertices of are and edges. We made partition of vertex set with respect to the connection number as . For this purpose, we draw graphs of for using Mathematica and compute degree sequence as . Degree sequence of implies , and are the only possibilities of the connection numbers associated to the vertices of and , and 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 Table 1 depict , and . By using this partition, we computed :Let , and from vertex partition , we obtainIn Table 2, we list our observation for the number of edges with similar end vertex connection number and generalize our findings for arbitrary value of .

The generalization based on Table 2 provides edge partitions as , and . Using this partition, we computed defined asLet :Now, we compute which is defined asLet and

##### 4.2. Results for

A. R. Ashrafi in [29] computed winner index of . In Theorem 2, we computed connection Zagreb indices of whose chemical graph for is shown in Figure 4.

Theorem 2. *The , and of are equal to*

*Proof. *The total number of vertices of are and edges . Like , we partitioned the vertex set as and the edge set as by using edge and vertex segment strategy along with edges and vertex listing technique for desired partition.

The computations in Table 3 depict , and . By using this partition, we compute asLet ; then, by using vertex set partition , we obtainIn Table 4, we list our observation for the number of edges with similar end vertex connection number and generalize our findings for .

The generalization based on Table 4 provides edge partitions as , and . Using this partition, we computed and :Let ; then, by using vertex edge partition , we obtainFor ,Let :

##### 4.3. Results for

In Theorem 3, we computed connection Zagreb indices of , for . for is shown in Figure 5.

Theorem 3. *The , and of are equal to*

*Proof. *The total number of vertices of are and edges. Similar technique is used as in and computation. We partitioned the vertex set as and the edge set as .

The computations in Table 5 give vertex partition , and . By using this partition, we compute :Let :In Table 6, we list our observation for the number of edges with similar end vertex connection number of and generalize for arbitrary value of .

The generalization based on Table 6 provides edge partitions as , and . Using this partition, we computed and :Let ,Now, for ,Let :

##### 4.4. Comparison of Results

In Figure 6, Figure 6(a) interprets behavior of , Figure 6(b) of , and Figure 6(c) of for .

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##### 4.5. Results for

Graph shown in Figure 7 is the associated chemical graph of for arbitrary value of . It consists of a k-gon as the central core cycle with layers of hexagons around the core.

Theorem 4. *The , and of are*

*Proof. *The chemical graph of has cycle of length corresponding to core of , where represent the number of layers of hexagons around the core. Total vertices of is and edges is [30]. For computation of for generalized nanocone , we used their edges and vertex partition. From the construction of , it is clear that the connection number associated to the vertices is except the vertices of outer layer of hexagons. So, the number of vertices with connection number is .

In the outer layer of hexagons, two vertices at each corner have connection number and all other have . So, and . Similarly, all the edges other than edges of the outer layer of hexagons have as end vertex connection number, i.e., . The edges in the outer layer of hexagons are partitioned as , and . We use this vertex and edge partition and technique as used in Theorems 1â€“3 for proof. Let and :Hence, the proof is completed.

Theorem 4 shows that connection Zagreb indices of increased by a factor (number of carbon atoms in the core). In Figure 8, Figures 8(a) and 8(b) interpret computed results of and , respectively, for any value of and 9 of .

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#### 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 for the physio-chemical properties of octane isomers: entropy, boiling point, density, enthalpies, acentric factor, etc. They concluded that has a correlation coefficient approximately and for acentric factor and entropy, respectively. Correlation of with physical properties was studied by Jakkannavar and Basavanagoud [33] and found good with enthalpy. Possibility of large-scale production of [34] and predictive ability of topological indices [35] encourages theoretical study of . We computed connection Zagreb indices , and of for in Theorems 1â€“3 and generalized our finding up to large value of in 4.4. These results facilitate topological characterization of which is helpful in finding new applications in the emerging field of nanotechnology.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors of this paper declare that they have no conflicts of interest.

#### Authorsâ€™ Contributions

All authors have contributed equally.

#### Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number no. R-2021-103.