Topological Study of 6.82 D Carbon Allotrope Structure

Abstract

Carbonallotropes are widely available and can be found in the atmosphere, the earth’s crust, and in living creatures in myriad forms. Allotropes are also used in several fields, including for medicinal and biological applications, due to their intriguing properties such as low resistance, high electron mobility, abnormal quantum hall effect, unconventional superconductivity in graphene, and so on. The theoretical analysis of carbon allotropes can hence be quite useful as it leads to a better understanding of the nature and behavior of these ubiquitous materials and also opens the door for even better applications. The objective of this research is to theoretically analyze the ${6.8}^2$ carbon allotrope by using four kinds of vertex degree based (VDB) topological indices (Tis), namely VDB multiplicative topological indices, VDB indices using M-Polynomial, VDB entropy measures, and irregularity indices. This analysis will extend the current body of knowledge available for this allotrope and help future researchers in the synthesis of new allotropes.

1. Introduction

Mathematical chemistry is a subfield of theoretical chemistry that uses graph theory to analyze molecule structure. In other words, it ensures the properties of the chemistry of this structure by specifying the vertices, edges, and degree vertices graph of the molecular structure using mathematical methods. Many chemistry problems can be solved using these strategies. The theory of chemical graphs is a branch of mathematical chemistry; in fact, graph theory connects chemistry with mathematics, and it uses graph theory to answer many of mathematical chemistry’s most challenging issues. The topologic index, also known as a connection index, characterizes the chemical makeup of a substance based on its molecular structure [1,2,3].

Tis of massive chemical structures, such as metal organic frameworks, can be particularly valuable in both characterization and computing physicochemical parameters that are difficult to compute for such vast networks in reticular chemistry [4,5,6]. In recent years, the synthesis of innovative reticular metal-organic frameworks and networks in which covalent fibers are braided into crystals has become more essential. Topological indices are numerical representations of molecule structural properties produced by the application of graph-theoretical principles to vast networks of interest in mystifying chemistry. Several degree-based indices have been developed to test the properties of compounds and drugs, and they are widely used in chemical and pharmacy engineering. Topological indices can be thought of as a collection of parameters on a molecular graph. This is crucial in theoretical physics and pharmacology science.

Carbon is a relatively common element that can be found in the atmosphere, the universe’s core, and living things in many forms. Under ambient conditions, carbon’s capacity to form sp, sp², and sp³hybridised bonds results in the formation of diverse allotropes [7].Carbon atoms form a flat sheet known as graphene when they have no curve, while a positive curvature generates the soccer ball-like structure known as buckyballs. For decades, scientists have hypothesized that a third variety—a structure with negative curvature—should exist. In this study we discussed about one of the typical negatively curved carbon allotrope named as ${6.8}^2$ carbon allotrope structure [8].

6.8² D Carbon Allotrope

The ${6.8}^2$ carbon allotrope structure is typical negatively curved carbon allotrope and it can be obtained by the condensation of truncated-icosahedral C₆₀ molecules. Twelve atoms, or six “double” bonds, are taken away from each C₆₀ molecule in such a way that the remaining 48atoms, arranged as eight hexagons, maintain cubic symmetry. The truncated structure, thus obtained, is joined to six similar structures along the six cubic face directions, resulting in four eight sided rings at each juncture. This process results in an allotrope which has been predicted to possess remarkable stability, approximately 0.23 eV/atom more stable than C₆₀. In relative terms, the smallest molecular fullerene known to have the same relative energy has 180 carbon atoms [7,9,10,11]. The ${6.8}^2$ carbon allotrope was also predicted to be an insulator. The spongy structure of the allotrope, that is, the large ordered hollows, could host alkali metal ions, similar to naturally occurring zeolite structures. In 2012, Szeflerand Diudea represented the structure in graph-theoretical terms, using the Omega polynomial [12,13,14,15]. The relatively high stability of this structure, as well as its numerous possible applications, prompted this study to conduct a structural theoretical examination of the allotrope. The structure is investigated at the molecular level in this article utilizing vertex degree based (VDB) and related indices.

There are currently no studies in the literature that look at this structure from a topological perspective. As a result, this research is unique, and it will contribute to a deeper knowledge of the allotrope as well as more precise predictions of its physical and chemical properties. The findings in this paper could be used to comparable allotropes that are designed and synthesized in the future. The basic mathematical terminology were covered in Section 2 of this article. The methodology was discussed in Section 3. VDB multiplicative Tis, VDB indices utilizing M-polynomial, VDB entropy measures, and VDB irregularity indices for the 6.8² Carbon Allotrope are computed in part 4 using edge-partitioning techniques. Section 6 concludes the paper by doing numerical analysis on the computed data.

2. Mathematical Terminologies

In this paper, $\Gamma$ represents a connected graph, $V$ and $E$ refer to the vertex set and the edge set, respectively. The degree of a vertex ${deg}_{\Gamma }\left(𝓋\right)$ in a graph $\Gamma$ is the number of edges that are adjacent to that vertex $𝓋$ [16]. We used edge-partition approaches to generate VDB Multiplicative Tis, VDB indices utilizing M-polynomial, VDB entropy measures, and VDB irregularity indices for the Carbon Allotrope.

2.1. Multiplicative Topological Indices

Some multiplicative topological indices have been researched in recent years, for example, in [17,18]. In Nano carbon Kwun et al. they computed the Multiplicative Degree-Based Topological Indices of Silicon-Carbon Si2C3-I[p,q] and Si2C3-II[p,q] [19].Yousaf, S.A. et al. discussed Carbon Nanotubes using degree based Tis [20].

The multiplicative topological index [21] is represented as

$MTI\left(\Gamma \right)={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}F\left({deg}_{\Gamma }\left(𝓊\right),{deg}_{\Gamma }\left(𝓋\right)\right)$

where $\prod$ denotes the product of the terms $F\left({deg}_{\Gamma }\left(𝓊\right),{deg}_{\Gamma }\left(𝓋\right)\right)$. The multiplicative version of the Wiener index was the first topological index investigated [22].

The first and second version of multiplicative F-indices [23,24] are described as follows

$M{F}_{1}I\left(\Gamma \right)={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}{deg}_{\Gamma }{\left(𝓊\right)}^2\times {deg}_{\Gamma }{\left(𝓋\right)}^2$

and

$M{F}_{2}I\left(\Gamma \right)={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}{deg}_{\Gamma }{\left(𝓊\right)}^2\times {deg}_{\Gamma }{\left(𝓋\right)}^2$

The first multiplicative hyper-Zagreb index is described [25] as

$FMHZI\left(\Gamma \right)={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}{\left({deg}_{\Gamma }\left(𝓊\right)+{deg}_{\Gamma }\left(𝓋\right)\right)}^2$

The multiplicative Harmonic index of a graph is defined as

$MHI\left(\Gamma \right)={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}\left(\frac{2}{{deg}_{\Gamma }\left(𝓊\right)+{deg}_{\Gamma }\left(𝓋\right)}\right)$

The multiplicative sum connectivity and Randić indices are represented [26,27] as

$MSCI\left(\Gamma \right)={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}\frac{1}{\sqrt{{deg}_{\Gamma }\left(𝓊\right)+{deg}_{\Gamma }\left(𝓋\right)}}$

and

$MRI\left(\Gamma \right)={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}\frac{1}{\sqrt{{deg}_{\Gamma }\left(𝓊\right)\times {deg}_{\Gamma }\left(𝓋\right)}}$

Multiplicative ABC index, multiplicative GA index, and are defined as

$MABCI\left(\Gamma \right)={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}\sqrt{\frac{{deg}_{\Gamma }\left(𝓊\right)+{deg}_{\Gamma }\left(𝓋\right)-2}{{deg}_{\Gamma }\left(𝓊\right)\times {deg}_{\Gamma }\left(𝓋\right)}}$

and

$MGAI\left(\Gamma \right)={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}\frac{2\sqrt{{deg}_{\Gamma }\left(𝓊\right)\times {deg}_{\Gamma }\left(𝓋\right)}}{{deg}_{\Gamma }\left(𝓊\right)+{deg}_{\Gamma }\left(𝓋\right)}$

The multiplicative augmented Zagreb index is defined as

$MAZI\left(\Gamma \right) ={\displaystyle \prod }_{𝓊𝓋\in E\left(\Gamma \right)}{\left(\frac{{deg}_{\Gamma }\left(𝓊\right)\times {deg}_{\Gamma }\left(𝓋\right)}{{deg}_{\Gamma }\left(𝓊\right)+{deg}_{\Gamma }\left(𝓋\right)-2}\right)}^3$

2.2. VDB Indices Utilising M-Polynomial

There are currently many algebraic polynomials available in the literature that can be used to calculate distance-based Tis. Among them, the Hosoya polynomial has been the most widely employed, since several distance-based indices can be computed using a single polynomial. In 2015, Deutsch and Klavžar [28] made a similar breakthrough for VDB indices in the form of M-polynomial. Similar to the Hosaya polynomial, the M-polynomial can also be used in the computation of several VDB indices. JulietrajaandVenugopal studied the topological descriptors of coronoid systems and Benzenoid systems using M-polynomials [29,30]. Farkhanda et al., introduced some new degree based topological indices via M polynomial [31]. Rauf, A. et al., computed an algebraic polynomial based topological study of graphite carbon nitride (g^–) molecular structure [32]. Guangyu, L. et al., carried out an analysis of carbon nanotube and polycyclic aromatic nanostar molecular structures [33].

The M-polynomial of $\Gamma$ is defined as

$M\left(\Gamma ;𝓌,𝓀\right)={\displaystyle \sum }_{\delta \le i\le j\le \Delta }{m}_{ij}\left(\Gamma \right){𝓌}^i{𝓀}^j$

where $\delta =Min\left\{{deg}_{\Gamma }\left(𝓋\right) :𝓋\in V\left(\Gamma \right)\right\}; \Delta =Max\left\{{deg}_{\Gamma }\left(𝓋\right) :𝓋\in V\left(\Gamma \right)\right\},$ and $m_{ij}\left(\Gamma \right)$ is the edge $𝓋𝓊\in E\left(\Gamma \right)$ for which $\left\{{deg}_{\Gamma }\left(𝓋\right),{deg}_{\Gamma }\left(𝓊\right)\right\}=\left\{i,j\right\}$. Table 1 shows that the M-polynomial Topological Indices.

$D_{𝓌}\left(𝓯\left(𝓌,𝓀\right)\right)=𝓌\frac{\partial 𝓯\left(𝓌,𝓀\right)}{\partial 𝓌},D_{𝓀}\left(𝓯\left(𝓌,𝓀\right)\right)=𝓀\frac{\partial 𝓯\left(𝓌,𝓀\right)}{\partial 𝓀}$, $L_{𝓌}\left(𝓯\left(𝓌,𝓀\right)\right)=𝓯\left({𝓌}^2,𝓀\right)$, $L_{𝓀}\left(𝓯\left(𝓌,𝓀\right)\right)=𝓯\left(𝓌,{𝓀}^2\right)$

$S_{𝓌}\left(𝓯\left(𝓌,𝓀\right)\right)={{\displaystyle \int }}_0^{𝓌}\frac{𝓯\left(t,𝓀\right)}{t}dt$, $S_{𝓀}\left(𝓯\left(𝓌,𝓀\right)\right)={{\displaystyle \int }}_0^{𝓀}\frac{𝓯\left(𝓌,t\right)}{t}dt$, $J\left(𝓯\left(𝓌,𝓀\right)\right)=𝓯\left(𝓌,𝓌\right)$, $D_{𝓌}^{\frac{1}{2}}\left(𝓯\left(𝓌,𝓀\right)\right)=\sqrt{𝓌\frac{\partial 𝓯\left(𝓌,𝓀\right)}{\partial 𝓌}}·\sqrt{𝓯\left(𝓌,𝓀\right)}$, $D_{𝓀}^{\frac{1}{2}}\left(𝓯\left(𝓌,𝓀\right)\right)=\sqrt{𝓀\frac{\partial 𝓯\left(𝓌,𝓀\right)}{\partial 𝓀}}·\sqrt{𝓯\left(𝓌,𝓀\right)}$, $s_{𝓌}^{\frac{1}{2}}\left(𝓯\left(𝓌,𝓀\right)\right)=\sqrt{{{\displaystyle \int }}_0^{𝓌}\frac{𝓯\left(𝓌,𝓀\right)}{t}dt}·\sqrt{𝓯\left(𝓌,𝓀\right)}$, $s_{𝓀}^{\frac{1}{2}}\left(𝓯\left(𝓌,𝓀\right)\right)=\sqrt{{{\displaystyle \int }}_0^{𝓀}\frac{𝓯\left(𝓌,𝓀\right)}{t}dt}·\sqrt{𝓯\left(𝓌,𝓀\right)}$, $Q_{\alpha }\left(𝓯\left(𝓌,𝓀\right)\right)={𝓌}^{\alpha }𝓯\left(𝓌,𝓀\right), \alpha \ne 0$.

2.3. VDB Entropy Measures for 6.8² Carbon Allotrope

Let $n$ be the order of a graph of size $m$ and $\phi$ is some meaningful information function. The Shannons entropy [34] of a graph is depicted as

(1)$EN{T}_{\phi }\left(\Gamma \right)=-{\displaystyle \sum }_{l=1}^n\frac{\phi \left({deg}_{\Gamma }\left({𝓋}_l\right)\right)}{{{\displaystyle \sum }}_{m=1 }^n\phi ({deg}_{\Gamma }\left({𝓋}_m\right))}log\left[\frac{\phi ({deg}_{\Gamma }\left({𝓋}_l\right)}{{{\displaystyle \sum }}_{m=1 }^n\phi ({deg}_{\Gamma }\left({𝓋}_m\right))}\right]$

Let ${𝓋}_l\in V\left(\Gamma \right)$ and the degree of ${𝓋}_l$ is represented by the information function $\phi \left({𝓋}_l\right)$ that is, $\phi \left({𝓋}_l\right)=de{g}_{\Gamma }\left({𝓋}_l\right)$. Then the Equation (1) can be rewritten as

$EN{T}_{\phi }\left(\Gamma \right)=-{\displaystyle \sum }_{l=1}^n\frac{{deg}_{\Gamma }\left({𝓋}_l\right)}{{{\displaystyle \sum }}_{m=1 }^n{deg}_{\Gamma }\left({𝓋}_m\right)}log\left[\frac{{deg}_{\Gamma }\left({𝓋}_l\right)}{{{\displaystyle \sum }}_{m=1 }^n{deg}_{\Gamma }\left({𝓋}_m\right)}\right]$

The fundamental theorem of graph theory is represented as ${\displaystyle \sum }_{m=1}^n{deg}_{\Gamma }\left({𝓋}_m\right)=2E$. As a result, the above equation becomes

(2)$EN{T}_{\phi }\left(\Gamma \right)=log\left(2E\right)-\frac{1}{2E}log\left[{\displaystyle \sum }_{l=1}^n{\left({deg}_{\Gamma }\left({𝓋}_l\right)\right)}^{{deg}_{\Gamma }\left({𝓋}_l\right)}\right]$

Chen et al. [35] introduced the entropy measure of an edge-weighted graph. If $\Gamma =\left(V\left(\Gamma \right);E\left(\Gamma \right); \phi \left(𝓊𝓋\right)\right)$ is an edge-weighted graph, where