Quality testing of spectrum-based distance descriptors for polycyclic aromatic hydrocarbons with applications to carbon nanotubes and nanocones

1 Introduction

2 Spectrum-based distance descriptors

Mathematically, a graph $\mathrm{\Gamma }$ is nothing but an order pair $\mathrm{\Gamma }=(V,E)$ comprising a set $V=V(\mathrm{\Gamma })$ of points called vertices and connection $E(\mathrm{\Gamma })\subseteq \left(\genfrac{}{}{0ex}{}{V}{2}\right)$ between those pairs of points called edges. A chemical/molecular graph is a hydrogen-depleted chemical structure obtained from a chemical compound. Vertices (resp. edges) in a chemical graph correspond to atoms (resp. bonds) in the underlying chemical compound. Two vertices are called adjacent if there is an edge between. The number of adjacent vertices to a given vertices $x\in V(\mathrm{\Gamma })$ is called the degree or valency of x and is denoted by $d_x$ . A vertex degree in a chemical graph, corresponding to an organic chemical compound, can be at most four, since the valency of the carbon atom is four. For more notations and terminologies on chemical graph theory, we refer the reader to Diudea et al. (2001) and Gutman and Polansky (1986).

Topological descriptors are mathematical functions mapping simple connected graphs to real numbers having significant applications in chemistry (Balaban et al., 1983). Valency/degree-based topological descriptors possess defining structures based on valencies of vertices, whereas, the defining structures of distance-based topological descriptors are based on distances between vertices in graphs. The atom-bond connectivity index, the Randić connectivity index, the sum-connectivity index and Zagreb indices, among others, are examples of valency-based descriptors. The Wiener index, the Balaban index, the Szeged index, among others, are included in the examples of distance-based descriptors. Spectrum-based topological descriptors comprise the defining structure based on eigenvalues of distance-related or valency-related chemical matrices. The adjacency energy, the Estrada index, the distance energy and Estrada index and Laplacian & signless Laplacian energies and Estrada indices, among many others, are included in the list of spectrum-based topological descriptors. Hosamani et al. (2017) reviewed various results on energies of different graph-related matrices. For a book on adjacency energy, we refer to Li et al. (2012). Moreover, for a book on the spectral radius of different graphical matrices, we refer to Stevanović (2014). Certain computational results on degree and distance-based topological descriptors are presented in Hayat (2017), Hayat and Imran (2014), Hayat et al. (2019b), and Hayat et al. (2020b).

Next we define some well-known distance-related chemical matrices. Later, spectrum-based distance descriptors are defined based on the eigenvalues of these matrices.

3 A unified computational method

In this section, we present a unified computational technique to compute all the spectrum-based topological descriptors in Section 2. The proposed technique is the main tool in computing these eigenvalues related topological descriptors for polycyclic aromatic compounds, carbon nanotubes and carbon nanocones in Sections 4, 5 & 6 respectively.

Among the other classes of topological descriptors such as valency-based descriptors, distance-based descriptors, counting related molecular descriptors, the class of eigenvalues-based topological descriptors is highly non-trivial and somewhat complicated since they are defined based on eigenvalues of graphical matrices. The branch of graph theory which derives certain structural properties of graphs from the eigenvalues of their related graph-theoretic matrices is called the spectral graph theory. Spectral graph theory assists in computing the eigenvalues of graphs on small order, say up to six vertices. There are also characterization results in spectral graph theory such as graphs with smallest eigenvalue at least $-2$ studied in Desai and Rao (1994). For more details on spectral graph theory, we refer the reader to the book by Cvetković et al. (1995) and by Brouwer and Haemers (2012). Although there is a rich literature available on graph spectra but current tools do not assist in computing all the eigenvalues of higher-order graphs. Thus, it is essential to develop computational methods based on computer softwares to compute the complete spectrum of eigenvalues of a given matrix which corresponds to a graph with many vertices.

Before we go into the details of our proposed method, we review all known methods available in the literature. Ashrafi (2010, 2009) proposed a computer-based computational technique to calculate the adjacency energy and adjacency Estrada index of a chemical graph i.e. graph based on organic chemical structures and therefore have maximum valency up to 4. Their computational method makes use of certain computer software packages such as HyperChem (2002) to draw a chemical graph, Diudea et al. (2002) to calculate its adjacency matrix and finally GAP (The GAP Team, 1992) to calculate the adjacency energy and adjacency Estrada index. Hayat (2017) proposed a computational technique for various distance-based topological descriptors just by using newGraph (Stevanović et al.) and MatLab (MATLAB). Furthermore, a comparative analysis in Hayat (2017) with Ashrafi (2010) and Ashrafi and Sadati (2009) was also conducted to show that the former possesses less computational complexity and is more diverse in computing distance-based topological descriptors. Hayat et al. (2019b) extended the method in Hayat (2017) to the class of degree-distance-based and degree-based descriptors. Hayat et al. (2019a) presented a similar technique to compute the adjacency energy, the adjacency Estrada index and the inertia indices. In this paper, we extend the technique of Hayat et al. (2019a) to the 33 spectrum-based topological descriptors in Section 2.”In what follows, the method by Ashrafi (2010, 2009) is explained, for a chemical graph H, in a three-step algorithm:

Step 1: Draw H on HyperChem to obtain.hin file corresponding to H.

Step 2: Input the.hin file to Topocluj and calculate its adjacency matrix A. Topocluj will output the respective.m file.

Step 3: Insert the.m file in a GAP program developed by Ashrafi (2010) and Ashrafi and Sadati (2009) to calculate the A-energy and A-Estrada index of H.”.

Fig. 1 presents a flowchart of the method by Ashrafi et al.

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Fig. 1

Flowchart of the technique by Ashrafi et al.

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Although this method by Ashrafi et al. provides a platform to perform computational eigenvalue analysis for chemical graphs, it has certain disadvantages. For instance, it is restricted to chemical graphs which makes it not applicable for inorganic structures. Moreover, it is restricted to calculate the A-energy and A-Estrada index. We fill these gaps by proposing a unified computational method which can compute any spectrum-based descriptor from Section 2 while possesses less computational & algorithmic complexities. Rather than using HyperChem and Topolocluj for drawing a chemical graph and calculate its adjacency matrix, we use a single and more efficient platform i.e. newGraph (Stevanović et al.) which ”is a fully integrated environment used for improving a research process in graph theory. It is compatible both with conveniently drawing a general graph and then computing various graph-theoretic parameters including the adjacency and distance matrices. For a general graph G, our two-step computational method is explained as follows:”.

Step 1: ”Use newGraph as single platform to both draw G and calculate its distance matrix A.

Step 2: Insert the matrix A into our MatLab (MATLAB) program and compute any of the spectrum-based descriptor from Section 2.”.

Fig. 2 depicts a flowchart of our proposed method.

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Fig. 2

Flowchart of our proposed method.

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We have created a public open repository on GitHub where we have uploaded an early view of ReadMe file, our MatLab code and a PDF file to explain the work- ing pattern of our technique with a minimal working example (MWE). The GutHub page is accessible via https://github.com/Sakander/Distance-based-spectral-descriptors.git.

A comparative analysis has been conducted to measure the efficiency of our method with the method by Ashrafi et al. We fix two benzenoid hydrocarbons i.e. coronene and ovalene (see Fig. 3) for our comparative analysis. Computational complexity has been investigated for these two graphs and the data is provided in Table 1.

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Fig. 3

The graphs of coronene and ovalene.

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We also conduct comparative analysis between newGraph and Topolocluj & HyperChem in our method and Ashrafi and coauthors, respectively. Table 2 gives a detailed comparison between these two software packages and exhibits their pros and cons. Collectively Tables 1 & 2 provide a detailed comparative analysis between our method and the technique by Ashrafi et al. These tables shows that our proposed computational method is less computationally & algorithmically complex and more diverse in calculating spectrum-based topological descriptors.

4 Predictive potential of spectrum-based distance descriptors

To investigate the correlation power and to determine the efficiency in quantitative structure activity/property relationship studies of spectrum-based topological descriptors, this section conducts a comparative testing to do the needful.

Standard enthalpy/heat of formation and normal boiling point are the standard chemical & physical properties which are opted for any comparative testing of topological descriptors. Chemically speaking, the normal boiling point is selected to advocate the van-der-Waals and intermolecular type interactions, whereas, the heat of formation is selected to be the representative of thermal characteristics. These two physicochemical characteristics are then correlated with a chosen set of topological descriptors. Better the correlation, better the prediction potential of a descriptors, is the simple criterion to access the performance of a topological descriptor.

Although it is customary to choose isomeric alkanes to be considered as molecules for this sort of comparative testings, we have selected lower polycyclic aromatic hydrocarbons (PAHs). The reason is that isomeric alkanes are thought to represent acyclic molecular structures, however, PAHs are considered to represent both cyclic and acyclic chemical structures. Graph-theoretic reasoning behind this fact is that trees are considered subgraphs of general graphs having cycles. Another reason of choosing PAHs is that some spectrum-based topological descriptors are defined to generalize the previously defined descriptors from acyclic structures to cyclic. Note further that for a comparative testing the number of molecules must be sufficiently large enough to acquire reliable and authentic statistical inferences. Moreover, the experimental data should be publicly available conveniently. Our choice of 22 lower polycyclic aromatic hydrocarbons meet both of these standard requirements. Fig. 4 exhibit the 22 PAHs with their aromaticity considered in this paper.”For experimental data regarding the normal boiling points of 22 lower polycyclic aromatic hydrocarbons (PAHs), we consulted the standard NIST databases (NIST Standard Reference Database) to acquire it. Moreover, the seminal work by Allison and Burgess (2015) provides the experimental data for the heat/enthalpy of formation for these polycyclic aromatic hydrocarbons. The acquired experimental data has been confirmed by tallying with a similar data available in the paper by Nikolić et al. (1998). We also refer the reader to a work by Putz et al. (2013, 2015) for an interesting interaction between topological descriptors, QSAR studies and PAHs.”

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Fig. 4

The 22 polycyclic aromatic compounds (PAHs) considered for our comparative testing.

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We use our proposed computational technique from Section 3 to calculate all the distance-based spectral descriptors from Section 2 for the 22 lower polycyclic aromatic hydrocarbons given in Fig. 4. Afterwards, the correlation is established between a topological descriptor and the experimental data for the normal boiling point and enthalpy of formation for the 22 PAHs. Although it is always preferably tried to establish a curvilinear correlation, we could not develop a curvilinear correlation for any of the spectrum-based distance descriptor. Thus, we measure the prediction power of a topological descriptor by evaluating its correlation coefficients with the physicochemical properties.”The data for such correlation coefficients with the normal boiling point as well as with the standard enthalpy/heat of formation for lower benzenoid hydrocarbons is provided in Table 3, in which $\mathrm{\Delta }{H}_f^o$ (resp. b.p.) denotes the standard heat of formation (resp. the normal boiling point) of a molecular structure.”.”It is important to notice in Table 3 that, for a given spectrum-based distance descriptor, the two correlation coefficients with the standard enthalpy of formation and the normal boiling point are considerably close to each other. This ensures us the significant appropriateness of our chosen physicochemical properties, very distinctive in nature though. This also suggests that these two physicochemical characteristics could be chosen to investigate the efficiency of any topological or molecular descriptor, in general.”

4.1

4.1 Analysis of results

We have measured the efficiency of a topological descriptor in Table 3 by investigating the average of the two correlation coefficients. The simple criterion is that closer the average correlation coefficient to one, better the performance of topological descriptor is. Based on the average correlation coefficient, we have composed a list of top five best performing spectral descriptors. Table 4 gives the priority list for the five best spectrum-based distance descriptors. Table 5 provides a detailed statistical analysis for the prioritized five spectrum-based descriptors.

Table 4 The five best spectrum-based distance topological descriptors.

Topological descriptor	Position on the priority list
${\mathit{EE}}_{{\mathit{ABC}}^2}$	1
$E_H$	2
$E_{{\mathit{GA}}^2}$	3
${\mathit{EE}}_{{\mathit{GA}}^2}$	4
$E_{\mathit{DD}}$	5

Table 5 The regression model, determination coefficient, correlation coefficient and standard error of fit for the five best well-performing spectrum-based distance descriptors with 22 lower benzenoid hydrocarbons.

S.No	Index	Regression model	Statistics
1	${\mathit{EE}}_{{\mathit{ABC}}^2}$	$\mathit{BP}=-{127.05}_{\pm 25.88598}+{24.719}_{\pm 1.06243}{\mathit{EE}}_{{\mathit{ABC}}^2}$	$r^2=0.9916,\rho =0.9958,s=0.4885$
	${\mathit{EE}}_{{\mathit{ABC}}^2}$	$\mathrm{\Delta }{H}_f^o=-{49.606}_{\pm 44.58162}+{14.125}_{\pm 1.82975}{\mathit{EE}}_{{\mathit{ABC}}^2}$	$r^2=0.9284,\rho =0.9635,s=1.4248$
2	$H_E$	$\mathit{BP}=-{37.54}_{\pm 20.40207}+{19.614}_{\pm 0.77688}{H}_E$	$r^2=0.9928,\rho =0.9964,s=0.5681$
	$H_E$	$\mathrm{\Delta }{H}_f^o={4.4564}_{\pm 42.55145}+{11.093}_{\pm 1.62030}{H}_E$	$r^2=0.9107,\rho =0.9543,s=2.0066$
3	$E_{{\mathit{GA}}^2}$	$\mathit{BP}=-{50.033}_{\pm 24.39614}+{20.061}_{\pm 0.92836}{E}_{{\mathit{GA}}^2}$	$r^2=0.9951,\rho =0.9918,s=0.6473$
	$E_{{\mathit{GA}}^2}$	$\mathrm{\Delta }{H}_f^o=-{0.5631}_{\pm 47.16455}+{11.266}_{\pm 1.79477}{E}_{{\mathit{GA}}^2}$	$r^2=0.8955,\rho =0.9463,s=2.1192$
4	${\mathit{EE}}_{{\mathit{GA}}^2}$	$\mathit{BP}=-{33.94}_{\pm 24.79515}+{10.541}_{\pm 0.51096}{\mathit{EE}}_{{\mathit{GA}}^2}$	$r^2=0.9893,\rho =0.9946,s=1.2890$
	${\mathit{EE}}_{{\mathit{GA}}^2}$	$\mathrm{\Delta }{H}_f^o={9.169}_{\pm 46.93558}+{5.9051}_{\pm 0.96722}{\mathit{EE}}_{{\mathit{GA}}^2}$	$r^2=0.8902,\rho =0.9435,s=4.1320$
5	$E_{\mathit{DD}}$	$\mathit{BP}={4.0165}_{\pm 22.22913}+{3.7084}_{\pm 0.17360}{E}_{\mathit{DD}}$	$r^2=0.9900,\rho =0.9950,s=3.5415$
	$E_{\mathit{DD}}$	$\mathrm{\Delta }{H}_f^o={31.288}_{\pm 44.59168}+{2.0704}_{\pm 0.34825}{E}_{\mathit{DD}}$	$r^2=0.8849,\rho =0.9407,s=12.0303$

Results support our newly defined spectrum-based distance descriptors as the second atom-bond connectivity Estrada index gives the best performance of them all. Moreover, two more newly introduced spectral descriptors based on the second geometric-arithmetic matrix make it to the top-five priority list. Those are the second geometric-arithmetic energy and Estrada index of graphs. Among the spectral descriptors available in the literature, the Harary energy and the degree-distance energy also make it to the”priority list of the five best spectrum-based distance descriptors. The results warrant further usage of the second atom-bond connectivity energy and Harary energy in the structure–activity and structure–property models.”.

The data in Table 3 suggests a number of favorable and unfavorable conclusions. By favorable outcome, we mean some less-known spectrum-based distance descriptors such as the Harary energy and the degree-distance energy etc. performed extraordinarily, even though they are less-reputable among the researchers. On the other hand, some well-established and well-reputable spectrum-based distance descriptors such as the distance energy and Estrada index and the distance Laplacian & signless Laplacian energies and Estrada indices, could not perform up to their reputation among chemical graph theorists and theoretical chemists.

Practically speaking, topological descriptors having correlation coefficients less then 0.95 are not recommended for their usage in quantitative structure–activity and structure–property models. Having said that, the well-known spectrum-based descriptors such as the the distance energy and the distance Estrada index, the Laplacian & signless Laplacian energies and Estrada indices, the Szeged and Schultz energies and Estrada indices etc. have correlation coefficients lower than 0.95 which is considered very poor from application perspectives. They should not be suggested for the quantitative structure–activity and structure–property models.”For their use in structure–property relationship models, we prefer the newly defined second atom-bond connectivity Estrada index and the Harary energy which have both of the correlation coefficients grater than 0.95. Despite their popularity, the results warrant further usage of the second atom-bond connectivity Estrada index and the Harary energy in the quantitative structure–activity and structure–property models.””Next, we exhibit the scatter plots of best five spectrum-based distance descriptors with normal boiling point and enthalpy of formation for the selected 22 lower benzenoid hydrocarbons. Tables 6 and 7 depict these scatter plots and the corresponding linear regression models. This explains the data dispersion and the error between the exact and correlated data points.””In conclusion, for their use in structure–property relationship models, we prefer only the second atom-bond connectivity energy and the Harary energy which have both of the correlation coefficients grater than 0.95. Despite of their popularity, the results warrant further usage of the second atom-bond connectivity index in the structure–activity and structure–property models.”

Table 6 Scatter plots between top five spectrum-based distance descriptors and the normal boiling point of 22 lower PAHs.

Table 7 Scatter plots between top five spectrum-based distance descriptors and the enthalpy of formation of 22 lower PAHs.

5 Applications to carbon polyhex nanotubes

When the size of molecular structures reduces to the range 1–100 nm, the study of nanotechnology starts. Having potential applications in computer science, electronics and medicine, nanotechnology creates many new devices and materials. Carbon nanotubes (CNTs) are cylindrical nano-skeletons comprising allotropic carbon. Carbon nanotubes have been found to possess relatively more applicative potential than their other counterparts.

The family of carbon polyhex nanotubes are the ones whose cylindrical surface is encompasses by hexagonal tessellation. Carbon polyhex nanotubes possess extraordinary stability for their existence in nature and, therefore, CNTs have fascinating thermal, electrical and mechanical characteristics (Dresselhaus et al., 1996). These significantly distinctive features of carbon polyhex nanotubes make these nanotubes the most studied nanostructures so far. Based on chirality, carbon polyhex nanotubes consist of different shapes such as zigzag, armchair and chiral etc. Fig. 5 depicts the zigzag and armchair structures of carbon polyhex nanotubes.

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Fig. 5

3D view of zigzag and armchair polyhex nanotubes.

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In order to study spectrum-based topological descriptors, we construct 2D graphs of these zigzag and armchair carbon polyhex nanotubes. Denoted by ${\mathit{TUZC}}_6[m,n]$ , the zigzag polyhex nanotube is parameterized having m (resp. n) to be the number of hexagons in a row (resp. column). Similarly, the armchair polyhex nanotube is denoted by ${\mathit{TUAC}}_6[m,n]$ , where the parameter m (resp. n) is defined to be the number hexagons in a fixed row (resp. column). Just for notational convenience, we denote an ( $m,n$ )-dimensional zigzag (resp. armchair) nanotube by ${\mathit{ZC}}_6[m,n]$ (resp. ${\mathit{AC}}_6[m,n]$ ). Fig. 6 exhibits the ( $m,n$ )-dimensional zigzag and armchair polyhex nanotubes.

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Fig. 6

An ( $m,n$ )-dimensional zigzag and armchair polyhex nanotubes.

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In this section, we employ our computational technique from Section 3 to calculate five best spectrum-based distance descriptors in Table 4 for carbon polyhex nanotubes. We choose two infinite families of zigzag polyhex nanotubes i.e. ${\mathit{ZC}}_6[m,n]$ with $m=4$ and $m=5$ . The remaining cases could be dealt similarly. Table 8 shows the calculations obtained for the zigzag polyhex nanotube ${\mathit{ZC}}_6[4,n]$ for $2⩽n\le 10$ , as ${\mathit{ZC}}_6[4,1]$ is trivial.

Next we use the cftool of MatLab on data in Table 8. cftool is a tool for curves and surface fitting in MatLab. We use R2013a version of the MatLab in our approximations. Our MatLab program generated the following results with 95% confidence intervals of coefficients: $$\begin{array}{cc}\hfill {\mathit{EE}}_{{\mathit{ABC}}^2}({\mathit{ZC}}_6[4,n])=& {18.77}_{\pm 0.15}n+{7.214}_{\pm 1.003},\hfill \\ \hfill E_H({\mathit{ZC}}_6[4,n])=& {25.62}_{\pm 0.01}n-{1.711}_{\pm 0.07},\hfill \\ \hfill E_{{\mathit{GA}}^2}({\mathit{ZC}}_6[4,n])=& {26.01}_{\pm 0.35}n-{6.639}_{\pm 2.287},\hfill \\ \hfill {\mathit{EE}}_{{\mathit{GA}}^2}({\mathit{ZC}}_6[4,n]))=& {51.42}_{\pm 1.03}n-{20.2}_{\pm 6.73},\hfill \\ \hfill E_{\mathit{DD}}({\mathit{ZC}}_6[4,n])=& {144.1}_{-0.1}n-{30.51}_{\pm 0.31}\text{.}\hfill \end{array}$$ These results will help in correlating the $\pi$ -electronic energies of ${\mathit{ZC}}_6[4,n]$ with significantly higher accuracy.

We also use our computational method in Section 3 to compute the best five spectrum-based topological descriptors exhibited in Table 4 for ${\mathit{ZC}}_6[5,n]$ nanotube. Table 9 shows the calculations obtained for the zigzag polyhex nanotube ${\mathit{ZC}}_6[5,n]$ for $2⩽n\le 10$ , as ${\mathit{ZC}}_6[5,1]$ is trivial.

Next we use the cftool on data in Table 9 for curve fitting purpose. Our MatLab program generated the following results with 95% confidence intervals of coefficients: $$\begin{array}{cc}\hfill {\mathit{EE}}_{{\mathit{ABC}}^2}({\mathit{ZC}}_6[5,n])=& {22.74}_{\pm 0.15}n+{6.724}_{\pm 0.976},\hfill \\ \hfill E_H({\mathit{ZC}}_6[5,n])=& {31.53}_{\pm 0.01}n-{2.103}_{\pm 0.059},\hfill \\ \hfill E_{{\mathit{GA}}^2}({\mathit{ZC}}_6[5,n])=& {31.63}_{\pm 0.5}n-{7.174}_{\pm 3.25},\hfill \\ \hfill {\mathit{EE}}_{{\mathit{GA}}^2}({\mathit{ZC}}_6[5,n]))=& {62.42}_{\pm 1.43}n-{21.35}_{\pm 9.34},\hfill \\ \hfill E_{\mathit{DD}}({\mathit{ZC}}_6[5,n])=& {179.5}_{+0.1}n-{37.76}_{\pm 0.3}\text{.}\hfill \end{array}$$

These results will help in correlation the $\pi$ -electronic energies of ${\mathit{ZC}}_6[5,n]$ with significantly higher accuracy.

6 Applications to one-hexagonal carbon nanocones

Carbon nanostructures are among those organic structures which possess a fascinating phenomenon of the occurrence of hollow carbon structures. Other than nanotubes and fullerenes, carbon nanocones are observed as nano-caps at the ends of a nanotube. They could also be observed as a free-standing structure on a flatten-up graphite surface. Extraction of a fan from a graphite sheet, rolling the remaining sector about its apex and then joining the two open sides would normally generate a carbon nanocone structure. Especially, when the angle of apex becomes more than 60∘, a hexagon appears at the top of the apex of the nanocones. One-hexagonal nanocone is a carbon nanocone, which has a hexagon on the tip of its apex, called its core. Fig. 7 depicts two different 3D views of a one-hexagonal nanocone.

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Fig. 7

3D views of a one-hexagonal nanocone.

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We denote an n-dimensional one-hexagonal nanocone by ${\mathit{CNC}}_6[n]$ , where n is the number of hexagonal layers covering up the conical surface of the nanocone. Moreover, six in the subscript shows that the core of the nanocone is a hexagon. Fig. 8 depicts the 3-dimensional ${\mathit{CNC}}_6[3]$ nanocone.

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Fig. 8

The 3-dimensional ${\mathit{CNC}}_6[3]$ nanocone.

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We also use our computational method in Section 3 to compute the best five spectrum-based topological descriptors exhibited in Table 4 for one-hexagonal nanocone ${\mathit{CNC}}_6[n]$ . Table 10 shows the calculations obtained for the one-hexagonal nanocone ${\mathit{CNC}}_6[n]$ for $1⩽n\le 10$ .

By using R2013a version of the MatLab, we employ cftool on data in Table 10. Our MatLab program generated the following results with 95% confidence intervals of coefficients: $$\begin{array}{cc}\hfill {\mathit{EE}}_{{\mathit{ABC}}^2}({\mathit{CNC}}_6[n])=& {66.21}_{\pm 12.3}n-{128.3}_{\pm 76.35},\hfill \\ \hfill E_H({\mathit{CNC}}_6[n])=& {90}_{\pm 80.2}n-{61.38}_{\pm 497.38},\hfill \\ \hfill E_{{\mathit{GA}}^2}({\mathit{CNC}}_6[n])=& {97.46}_{\pm 18.54}n-{197.1}_{\pm 115.06},\hfill \\ \hfill {\mathit{EE}}_{{\mathit{GA}}^2}({\mathit{CNC}}_6[n]))=& {195.2}_{\pm 37.2}n-{398.4}_{\pm 230.8},\hfill \\ \hfill E_{\mathit{DD}}({\mathit{CNC}}_6[n])=& {561.5}_{\pm 109.4}n-{1168}_{\pm 678}\text{.}\hfill \end{array}$$ These results will help in correlation the $\pi$ -electronic energies of one-hexagonal nanocones with significantly higher accuracy.

7 Conclusion

Declarations

CRediT authorship contribution statement

Sakander Hayat: Conceptualization, Methodology, Software, Writing- reviewing and editing. Suliman Khan: Data curation, Investigation, Visualization, Writing- original draft preparation. Muhammad Imran: Supervision, Project administration, Funding acquisition.