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Abstract

A lithium metal anode has a high energy density, but its applications face numerous challenges, particularly the irregular deposition of lithium dendrites, which form due to the aggregation of lithium clusters, posing serious safety risks. Research has revealed that small lithium clusters have the potential to enhance the electrode potential in Lithium-ion batteries (LIBs), and cluster energy affects the stability of lithium clusters. Modern computational tools and DFT calculations of lithium clusters can provide invaluable insights into their chemical, physical, and structural properties. However, calculating structural properties through large-scale experiments can be time-consuming and expensive. To overcome the challenges inherent in studying lithium clusters, this study employs a novel approach. We construct molecular graphs for the most stable structures and leverage CoM Polynomial computations to generate codescriptors, enabling efficient and accurate predictions of cluster energy for diverse lithium cluster configurations. The curvilinear regression analysis filters out highly significant regression equations and highly predictive codescriptors. Additionally, we investigated two molecular structures of porous graphene, namely linear and triangular, and obtained topological codescriptors’ analytical expressions by utilizing the CoM Polynomial in each case. A deeper understanding of both lithium clusters and porous graphene properties is essential for optimizing LIBs’ performance and stability.

Keywords

Topological codescriptors lithium clusters cluster energy porous graphene

Introduction

In recent years, the remarkable physical and chemical properties of clusters have sparked significant interest, leading to rapid developments in interdisciplinary research.^1,2 Clusters act as an intermediate transition state in the transformation of substances from atoms and molecules to macroscopic objects, representing a novel material structure with diverse applications in fields like chemical sensing, optical and optoelectronics, fluorescence, biomedical, catalysis, solar cells, energy, and environment.^3–9 Of particular interest are small metal clusters, especially lithium clusters, which can form with various elements and appear in processes like electrodeposition and condensation. For instance, lithium clusters form on the anode of a lithium battery during charging, but they can also cause stability issues, such as the growth of lithium dendrites.^10,11 Earlier studies have shown that lithium cluster properties, like binding energy, cluster energy, and stability, are related to their size, with most research focusing on small clusters due to computational complexity limitations.^12,13 Density functional theory and other calculation methods have been used to study various properties of lithium clusters, including ground and excited state geometry, electronic structure, binding and dissociation energies, ionization potentials, total energy, and cluster energy for different cluster sizes.^14–19 However, these techniques are computationally expensive and time-consuming.

Research into better electrode materials for batteries has led to the discovery of carbonaceous nanomaterials. Among these, graphene stands out for its exceptional properties, including high specific surface area, excellent electrical conductivity, mechanical strength, and chemical stability.^20–23 However, graphene’s tendency to stack together inhibits its electrochemical performance. To overcome this drawback, researchers have explored porous graphene materials, which enable electrolytes to penetrate electrodes and interact with active materials efficiently, facilitating electron transfer and ion diffusion.²⁴ In 2009, Bieri et al.²⁵ created a porous graphene structure (Figure 1) by altering the graphene structure, and its electronic properties were subsequently studied in Du et al.,²⁶ leading to nano electronic applications being discovered.

Figure 1.

(a) Structure of a fraction of the porous graphene network and (b) STM image of porous graphene.

Graphene-based porous materials possess the advantages of both graphene and porous materials, have been designed and developed with high porosity over time to offer better performance, and have unique properties that enable them to be used for a variety of applications such as its high surface area facilitates efficient adsorption, enabling applications in water purification,²⁷ catalysis,²⁸ and sensing.²⁹ Tunable pore sizes enable selective molecule capture, which is particularly important in gas separation.³⁰ Additionally, its excellent electrical conductivity supports electron transfer in super capacitors.³¹ Porous graphene is discussed in detail in Huang et al.³² and Zhang et al.³³

Lithium cluster properties are crucial for advancing battery technologies, and understanding their structural behavior also provides valuable insight into fundamental areas such as cluster formation and stability and nanomaterials design. Similarly the wide range of applications of porous graphene structures, with significant potential for advancements, necessitates studying their properties. Using quantitative structure-property relationships (QSPR) as a computational tool, researchers can predict nanostructure properties solely based on their chemical structure and molecular descriptors, enabling them to design nanostructures that have specific properties without having to actually synthesize them.³⁴ Chemical graph theory employs molecular graphs to represent chemical structures, with vertices symbolizing atoms and edges as bonds and topological molecular descriptors transform these graphs into mathematical real numbers, facilitating the prediction of properties for chemical structures.^35,36 Wiener’s³⁷ pioneering work introduced the Wiener index, the first topological molecular descriptor, enabling the prediction of boiling points for alkenes. This marked a significant leap in understanding the relationship between molecular structure and properties. Since then, numerous individual topological descriptors have been developed.³⁸ Notably, in 2015, Deutsch and Klavžar³⁹ developed the M-Polynomial technique, a powerful tool that provides numerous degree-based topological descriptors at once. Building upon the success of the M-Polynomial technique, the field of chemical graph theory has recently witnessed the introduction of two exciting new tools: the NM-Polynomial⁴⁰ and the CoM Polynomial.⁴¹

In Shanmukha et al.,^42,43 the computation of M Polynomial and NM Polynomial for porous graphene was conducted. Subsequently, in Govardhan and Roy,⁴⁴ adjustments were made to the edge partition by calculating the entropy of porous graphene. Some other graph property-based topological descriptor were computed for polyphenylene honeycomb structure in Krishnan and Rajan,⁴⁵ Krishnan et al.,⁴⁶ and Aftab,⁴⁷ and for porous nanographenes in Balasubramanian.⁴⁸ The structural and electronic properties of Lithium clusters up to ten were computed using density functional theory in Chetri et al.,⁴⁹ while the M-Polynomial computation of these clusters can be found in Chetri et al.⁵⁰

This article presents a twofold objective. Firstly, we propose a novel, computationally efficient, and alternate method for calculating the cluster energy of Lithium clusters. Secondly, we aim to address the gap in current literature by computing the CoM Polynomial of various porous graphene structures. This combined investigation is motivated by the significant role of both Lithium clusters and porous graphene in energy storage devices. Section “Preliminaries” lays the groundwork in graph theory, providing the necessary background for understanding the results presented in subsequent sections. Section “Relationship between the cluster energy with various topological codescriptor of lithium clusters” explores the relationship between specific Lithium cluster properties and the topological codescriptors obtained from the CoM Polynomial. Finally, Section “CoM polynomial of 2D porous graphene” presents detailed analytical results for linear and triangular porous graphene structures. This section includes both the computation of the CoM Polynomial and numerical comparisons in each case.

This article aims to develop a new method for calculating the cluster energy of lithium clusters and compute the CoM Polynomial for porous graphene structures. The motivation comes from the importance of lithium clusters and porous graphene in energy storage devices. The article starts with a background in graph theory, then explores the relationship between lithium cluster properties and topological codescriptors from the CoM Polynomial. Finally, it presents analytical results for linear and triangular porous graphene structures, including the computation of the CoM Polynomial and numerical comparisons (Figure 2). This work combines quantum chemistry and graph theory to advance battery technology.

Figure 2.

Overview of method used in this manuscript.

Preliminaries

In this article, $G (MG)$ represented as molecular graph with $| V (MG) |$ and $| E (MG) |$ as number of vertices and edges in a molecular graph. Two vertices are called adjacent if they have an edge in between them and $d (u)$ denotes degree of a vertex as number of vertices adjacent to vertex $u \in V (MG)$ . The complement of $G (MG)$ is denoted as $\bar{G} (MG)$ such that $uv \in \bar{G} (MG)$ if $uv \notin G (MG)$ . The CoM Polynomial for a simple connected graph G is defined as

CoM (G; x, y) = \bar{M} (G; x, y) = \sum_{i \leq j} {\bar{m}}_{ij} (G) x^{i} y^{j}

(1)

where

n_{i} = | V_{i} | for V_{i} = {v \in V (G) | d_{G} (v) = i},

m_{ij} = | E_{ij} | for E_{ij} = {uv \in E (G) | d_{G} (u) = i and d_{G} (v) = j}

{\bar{m}}_{ij} = | {\bar{E}}_{ij} | for {\bar{E}}_{ij} = {uv \in E (\bar{G}) | d_{G} (u) = i and d_{G} (v) = j}

and from Berhe and Wang⁵¹ ${\bar{m}}_{ij}$ can be find as

{\bar{m}}_{ij} = | {\bar{E}}_{ij} | = {\begin{matrix} \frac{n_{i} (n_{i} - 1)}{2} - m_{ii} for i = j \\ n_{i} n_{j -} m_{ij} for i < j \end{matrix}

(2)

From (1) several degree-based topological codescriptor (TCD) as shown in Table 1 can be obtained

Table 1.

List of topological codescriptor with notation obtained from CoM polynomial.

Topological codescriptor	Notation	Formula $g (d (u), d (v))$	Derivation from $f (x, y) = CoM (G; x, y)$
First Zagreb codescriptor	${\bar{M}}_{1} (G)$	$\sum_{uv \notin E (G)} (d (u) + d (v))$	$(D_{x} + D_{y}) (f (x, y))_{x = y = 1}$
Second Zagreb codescriptor	${\bar{M}}_{2} (G)$	$\sum_{uv \notin E (G)} d (u) d (v)$	$(D_{x} D_{y}) {(f (x, y))}_{x = y = 1}$
Second modified Zagreb codescriptor	$m {\bar{M}}_{2} (G)$	$\sum_{uv \notin E (G)} \frac{1}{d (u) d (v)}$	$(S_{x} S_{y}) {(f (x, y))}_{x = y = 1}$
Redefined third Zagreb codescriptor	$\bar{ReZ G_{3}} (G)$	$\sum_{uv \notin E (G)} d (u) d (v) (d (u) + d (v))$	$D_{x} D_{y} (D_{x} + D_{y}) {(f (x, y))}_{x = y = 1}$
Forgotten topological codescriptor	$\bar{F} (G)$	$\sum_{uv \notin E (G)} (d^{2} (u) + d^{2} (v))$	$({D_{x}}^{2} + {D_{y}}^{2}) {(f (x, y))}_{x = y = 1}$
Randić codescriptor	${\bar{R}}_{k} (G)$	$\sum_{uv \notin E (G)} (d (u) d (v))^{k}$	$({D_{x}}^{k} {D_{y}}^{k}) {(f (x, y))}_{x = y = 1}$
Inverse Randić codescriptor	${\bar{RR}}_{k} (G)$	$\sum_{uv \notin E (G)} \frac{1}{{(d (u) d (v))}^{k}}$	$({S_{x}}^{k} {S_{y}}^{k}) {(f (x, y))}_{x = y = 1}$
Symmetric division codescriptor	$\bar{SDD}$ (G)	$\sum_{uv \notin E (G)} \frac{χ^{2} (u) + χ^{2} (v)}{d (u) d (v)}$	$(D_{x} S_{y} + S_{x} D_{y}) (f (x, y))_{x = y = 1}$
Harmonic codescriptor	$\bar{H} (G)$	$\sum_{uv \notin E (G)} \frac{2}{d (u) + d (v)}$	$(2 S_{x} J) {(f (x, y))}_{x = 1}$
Inverse sum indeg codescriptor	$\bar{I} (G)$	$\sum_{uv \notin E (G)} \frac{d (u) d (v)}{(d (u) + d (v))}$	$(S_{x} J D_{x} D_{y}) {(f (x, y))}_{x = 1}$
Augmented Zagreb codescriptor	$\bar{A} (G)$	$\sum_{uv \notin E (G)} {\frac{d (u) d (v)}{d (u) + d (v) - 2}}^{3}$	$({S_{x}}^{3} Q_{- 2} J {D_{x}}^{3} {D_{y}}^{3}) {(f (x, y))}_{x = 1}$
Where $D_{x} = x (\frac{\partial (f (x, y))}{\partial x}), D_{y} = y (\frac{\partial (f (x, y))}{\partial y}), S_{x} = \int_{0}^{x} \frac{f (t, y)}{t} dt, S_{y} = \int_{0}^{y} \frac{f (x, t)}{t} dt, J (f (x, y)) = f (x, x), Q_{k} (f (x, y)) = x^{k} f (x, y)$

In Section “Relationship between the cluster energy with various topological codescriptor of lithium clusters” and “CoM polynomial of 2D porous graphene” the molecular graph of Lithium cluster is denoted as G $(L i_{n})$ . Moreover the molecular graph of Linear Porous Graphene and Triangular Porous Graphene is denoted as G $(LPG)$ and G $(TPG)$ respectively.

Relationship between the cluster energy with various topological codescriptor of lithium clusters

Molecular descriptors are crucial for predicting material features and designing economically, enabling rapid material discovery, saving time and costs compared to other techniques. This approach is particularly valuable in early material design stages, helping researchers focus on promising candidates and allowing to optimize conditions to enhance the likelihood of obtaining materials with desired properties. This section emphasizes an investigation conducted to assess the predictive capacity of codescriptors derived from CoM Polynomial. The inquiry aims to ascertain whether these codescriptors hold significance in predicting properties and whether they merit utilization in various applications. To achieve this we have constructed the molecular graph from the most stable structures of Lithium clusters^49,52 as shown below in Figure 3.

Figure 3.

Molecular graph of $L i_{n} (n = 4 - 10)$ with their $CoM (L i_{n}; x, y)$ .

The CoM Polynomial of the molecular graphs (Figure 3) was computed directly by employing the edge partition method. For example, the CoM Polynomial for the molecular graph of Lithium cluster $G (L i_{9})$ as shown in Figure 3, was obtained by the following way: $G (L i_{9})$ has $| E (L i_{9}) | = 16, | V (L i_{9}) | = 9$ .

This section investigates the predictive power of CoM Polynomial-derived codescriptors for material properties. Molecular graphs of stable Lithium clusters were constructed and their CoM Polynomials computed. The approach aims to determine if these codescriptors can predict material properties, enabling rapid material discovery and optimization.

The edge set of $G (L i_{9})$ can be classified into five classes as

$m_{23} = | E_{23} | = 0, m_{24} = | E_{24} | = 2, m_{34} = | E_{34} | = 8, m_{33} = | E_{33} | = 3, m_{44} = | E_{44} | = 3 .$ Similarly, $G (L i_{9})$ vertex set can be classified as

n_{2} = | V_{2} | = 1, n_{3} = | V_{3} | = 4, n_{4} = | V_{4} | = 4 .

Using 2, we have ${\bar{m}}_{23} = n_{2} n_{3} - m_{23} = 4$ , ${\bar{m}}_{24} = n_{2} n_{4} - m_{24} = 2$ , ${\bar{m}}_{34} = n_{3} n_{4} - m_{34} = 8$ and ${\bar{m}}_{33} = \frac{(n_{3}) (n_{3} - 1)}{2} - m_{33} = 3, {\bar{m}}_{44} = \frac{(n_{4}) (n_{4} - 1)}{2} - m_{44} = 3$ .

By the definition of CoM Polynomial

CoM (G; x, y) = \sum_{i \leq j} {\bar{m}}_{ij} (G) x^{i} y^{j}

\begin{matrix} CoM (L i_{9}; x, y) = \sum_{2 \leq 3} {\bar{m}}_{23} x^{2} y^{3} + \sum_{2 \leq 4} {\bar{m}}_{24} x^{2} y^{4} \\ + \sum_{3 \leq 3} {\bar{m}}_{33} x^{3} y^{3} + \sum_{3 \leq 4} {\bar{m}}_{34} x^{3} y^{4} \\ + \sum_{4 \leq 4} {\bar{m}}_{44} x^{4} y^{4} \end{matrix}

CoM (L i_{9}; x, y) = 4 x^{2} y^{3} + 2 x^{2} y^{4} + 3 x^{3} y^{3} + 8 x^{3} y^{4} + 3 x^{4} y^{4}

Below Table 2 lists ten codescriptor such as ${\bar{M}}_{1} (Li), {\bar{M}}_{2} (Li), m {\bar{M}}_{2} (Li), \bar{ReZ G_{3}} (Li), \bar{F} (Li), {\bar{R}}_{- \frac{1}{2}} (Li), \bar{SDD}$ ( $Li$ ), $\bar{H} (Li), \bar{I} (Li)$ , and $\bar{A} (Li)$ . These codescriptor were obtained from each of CoM Polynomial of the Lithium clusters. For example we have CoM Polynomial of $G (L i_{9})$ as

\begin{matrix} f (x, y) = CoM (L i_{9}; x, y) = 4 x^{2} y^{3} + 2 x^{2} y^{4} \\ + 3 x^{3} y^{3} + 8 x^{3} y^{4} + 3 x^{4} y^{4} \end{matrix}

then

$D_{x} f (x, y) = 8 x^{2} y^{3} + 4 x^{2} y^{4} + 9 x^{3} y^{3} + 24 x^{3} y^{4} + 12 x^{4} y^{4}$ .

$D_{y} f (x, y) = 12 x^{2} y^{3} + 8 x^{2} y^{4} + 9 x^{3} y^{3} + 32 x^{3} y^{4} + 12 x^{4} y^{4}$ .

$(D_{x} + D_{y}) (f (x, y)) = 20 x^{2} y^{3} + 12 x^{2} y^{4} + 18 x^{3} y^{3} + 56 x^{3} y^{4} + 24 x^{4} y^{4}$ .

$D_{y} D_{x} (f (x, y)) = 24 x^{2} y^{3} + 16 x^{2} y^{4} + 27 x^{3} y^{3} + 96 x^{3} y^{4} + 48 x^{4} y^{4}$ .

$(D_{x}^{2} + D_{y}^{2}) (f (x, y)) = 52 x^{2} y^{3} + 40 x^{2} y^{4} + 54 x^{3} y^{3} + 190 x^{3} y^{4} + 96 x^{4} y^{4}$ .

${D_{x}}^{k} {D_{y}}^{k} (f (x, y)) = (2^{k} 3^{k}) 4 x^{2} y^{3} + (2^{k} 4^{k}) 2 x^{2} y^{4} + (3^{k} 3^{k}) 3 x^{3} y^{3} + (3^{k} 4^{k}) 8 x^{3} y^{4} + (4^{k} 4^{k}) 3 x^{4} y^{4}$ .

$D_{x} D_{y} (D_{x} + D_{y}) (f (x, y) = 120 x^{2} y^{3} + 96 x^{2} y^{4} + 162 x^{3} y^{3} + 672 x^{3} y^{4} + 384 x^{4} y^{4}$ .

$S_{x} S_{y} (f (x, y)) = 4 \frac{1}{(2) (3)} x^{2} y^{3} + 2 \frac{1}{(2) (4)} x^{2} y^{4} + 3 \frac{1}{(3) (3)} x^{3} y^{3} + 8 \frac{1}{(3) (4)} x^{3} y^{4} + 3 \frac{1}{(4) (4)} x^{4} y^{4}$ .

$(S_{y} D_{x} + S_{x} D_{y}) (f (x, y)) = \frac{26}{3} x^{2} y^{3} + 5 x^{2} y^{4} + 6 x^{3} y^{3} + \frac{50}{3} x^{3} y^{4} + 6 x^{4} y^{4}$ .

$S_{x} J D_{y} D_{x} (f (x, y)) = \frac{24}{5} x^{5} + \frac{43}{6} x^{6} + \frac{96}{7} x^{7} + \frac{48}{8} x^{8}$ .

${S_{x}}^{3} Q_{- 2} J {D_{x}}^{3} {D_{y}}^{3} (f (x, y)) = 32 x^{3} + 16 x^{4} + \frac{2187}{64} x^{4} + \frac{13, 824}{125} x^{5} + \frac{12, 288}{216} x^{6}$ .

Table 2.

Lithium clusters $L i_{n} (n = 4 - 10)$ with their cluster energy and codescriptor obtained from CoM polynomial.

$L i_{n}$	Cluster energy	${\bar{M}}_{1} (Li)$	${\bar{M}}_{2} (Li)$	$m {\bar{M}}_{2} (Li)$	$\bar{ReZ G_{3}} (Li)$	$\bar{F} (Li)$	${\bar{R}}_{- \frac{1}{2}} (Li)$	$\bar{SDD}$ ( $Li$ )	$\bar{H} (Li)$	$\bar{I} (Li)$	$\bar{A} (Li)$
$L i_{4}$	2.65364	4	4	0.25	16	8	0.5	2	0.5	1	8
$L i_{5}$	3.57661	16	16	1	64	32	2	8	2	4	32
$L i_{6}$	4.77766	24	24	1.5	96	48	3	12	3	6	48
$L i_{7}$	5.99666	58	81	1.3611	490	182	3.6414	22.5	3.5381	13.662	95.039
$L i_{8}$	7.07009	88	120	2.6250	736	272	6.3284	36	6.1667	20.667	149.93
$L i_{9}$	7.99537	130	211	2.1042	1434	442	6.3995	42.33	6.3024	31.681	249.65
$L i_{10}$	9.06051	200	476	0.9425	4568	960	4.4389	42.4	4.4278	49.778	571.76

By the use of Table 1, we get

${\bar{M}}_{1} (L i_{9}) = {(D_{x} + D_{y}) f (x, y) |}_{x = y = 1} = 130$

${\bar{M}}_{2} (L i_{9}) = {D_{x} D_{y} f (x, y) |}_{x = y = 1} = 211$

$m {\bar{M}}_{2} (L i_{9}) = {S_{x} S_{y} f (x, y) |}_{x = y = 1} = 2.1042$

$\bar{ReZ G_{3}} (L i_{9}) = {D_{x} D_{y} (D_{x} + D_{y}) f (x, y) |}_{x = y = 1} = 1434$

$\bar{F} (L i_{9}) = {(D_{x}^{2} + D_{y}^{2}) (f (x, y)) |}_{x = y = 1} = 442$

${\bar{R}}_{- 1 / 2} (L i_{9}) = {{D_{x}}^{- 1 / 2} {D_{y}}^{- 1 / 2} f (x, y) |}_{x = y = 1} = 6.3995$

$\bar{SDD} (L i_{9}) = {(S_{y} D_{x} + S_{x} D_{y}) f (x, y) |}_{x = y = 1} = 42.33$

$\bar{H} (L i_{9}) = {2 S_{x} Jf (x, y) |}_{x = y = 1} = 6.3024$

$\bar{I} (L i_{9}) = {S_{x} J D_{x} D_{y} f (x, y) |}_{x = y = 1} = 31.681$

$\bar{A} (L i_{9}) = {{S_{x}}^{3} Q_{- 2} J {D_{x}}^{3} {D_{y}}^{3} f (x, y) |}_{x = y = 1} = 249.65$

The cluster energy (CE) of a Lithium cluster required to assemble a specific number of lithium atoms into a cluster. Table 2 listed the values of CE of $L i_{n} (n = 4 - 10)$ . To predict the CE of $L i_{n} (n = 4 - 10)$ we determine the best possible linear, quadratic, and cubic regression models. The linear, quadratic, and cubic regression models are often known as curvilinear regression analysis, which is what we examined. So, we examined the following equation.

P_{E} = \sum_{k = 1}^{3} α_{k} (CoD) + β

(3)

Equation (3) is the general form of the regression model, where $P_{E}$ represents Energy of Property such as of CE and $α_{k}$ represent coincides coefficients with $k = 1, 2 & 3$ and $β$ represents regression model constant. The predictive models developed in this section were obtained using the SPSS software, and their reliability was assessed using the values of the F-test, standard error (SE), and square of correlation coefficient $(R^{2})$ . Now, by examining the regression models described by equation (3) concerning the Cluster Energy feature and considering the codescriptor specific values outlined in Table 2, linear, quadratic, and cubic regression models are analyzed.

The findings in Tables 3 and 4 indicate that the Symmetric Division codescriptor $(\bar{SDD})$ emerges as the most effective descriptor for predicting the Cluster Energy of Lithium clusters when employing a linear regression model. In contrast, the First Zagreb codescriptor $({\bar{M}}_{1})$ proves to be the optimal descriptor for the quadratic regression model, and the Augmented Zagreb codescriptor $(\bar{A}$ ) stands out as the best descriptor in the cubic regression model.

CE = 2.655 + 0.136 (\bar{SDD} (Li))

(4)

CE = 2.757 + 0.063 ({\bar{M}}_{1} (Li)) - 1.62329 E - 4 {({\bar{M}}_{1} (Li))}^{2}

(5)

\begin{matrix} CE = 2.39992 + 0.05058 (\bar{A} (Li)) - 1.4799 E \\ - 4 {(\bar{A} (Li))}^{2} + 1.3975 E - 7 {(\bar{A} (Li))}^{3} \end{matrix}

(6)

Table 3.

Square of correlation coefficient $(R^{2})$ between cluster energy of lithium clusters and codescriptor obtained by linear, quadratic or cubic regression.

Cluster energy	${\bar{M}}_{1} (Li)$	${\bar{M}}_{2} (Li)$	$m {\bar{M}}_{2} (Li)$	$\bar{ReZ G_{3}} (Li)$	$\bar{F} (Li)$	${\bar{R}}_{- \frac{1}{2}} (Li)$	$\bar{SDD}$ ( $Li$ )	$\bar{H} (Li)$	$\bar{I} (Li)$	$\bar{A} (Li)$
Linear	0.90339	0.74831	0.15666	0.63594	0.77451	0.42123	0.96253	0.42094	0.88974	0.74463
Quadratic	0.982	0.95453	0.34771	0.92881	0.95948	0.4322	0.96512	0.42094	0.98201	0.97161
Cubic	0.9913	0.97835	0.48225	0.96628	0.97383	0.47133	0.97537	0.45242	0.99302	0.99486

Table 4.

Statistically predicted regression models that give best estimate for cluster energy.

Regression model	Curve equation	Adj $(R^{2})$	F	SE	SF
Linear	(4)	0.955	128.447	0.497	0.000
Quadratic	(5)	0.973	09.135	0.385	0.000
Cubic	(6)	0.990	204.550	0.231	0.001

Figure 4 shows plots of the linear, quadratic, and cubic regression equations (4), (5), and (6) of the Cluster energy with $\bar{SDD}$ ( $Li$ ), ${\bar{M}}_{1} (Li)$ , and $\bar{A} (Li)$ respectively.

Figure 4.

Plot of: (a) linear, (b) quadratic, and (c) cubic regression equations for the cluster energy of lithium clusters predicted by best codescriptor.

Equations (4), (5), and (6) can be used to calculate the CE of Lithium clusters by using codescriptor such as $\bar{SDD}$ , ${\bar{M}}_{1}$ , $\bar{A}$ with limited values upto 10 as can be seen in Table 2. We have therefore discovered the relationship between $\bar{SDD}$ , ${\bar{M}}_{1}$ , $\bar{A}$ and Lithium clusters in order to determine CE for any number of Li atoms. Based on the values of the correlation coefficient $(R^{2})$ in linear, quadratic, and cubic regression models respectively, we have obtained following equations.

\bar{SDD} = - 29.861 (\pm 5.157) + 7.638 (\pm 0.708) n

(7)

\begin{matrix} {\bar{M}}_{1} = 96.429 (\pm 29.96) - 43.905 (\pm 9.04) n \\ + 5.381 (\pm 0.641) n^{2} \end{matrix}

(8)

\begin{matrix} \bar{A} = - 1355.181 (\pm 578.566) + 708.369 (\pm 270.434) n \\ - 119.523 (\pm 40.162) n^{2} + 6.783 (\pm 1.907) n^{3} \end{matrix}

(9)

Figure 5 shows plots of the linear, quadratic and cubic regression equations (7), (8) and (9) between $\bar{SDD}$ , ${\bar{M}}_{1}$ , $\bar{A}$ and number of Li atoms, respectively.

Figure 5.

Plots of the linear, quadratic, and cubic regression equations (7), (8), and (9) between $\bar{SDD}$ , ${\bar{M}}_{1}$ , $\bar{A}$ , and number of Li atoms, respectively.

Using equations (7), (8), and (9) for any number of Li atoms in a Lithium clusters one can find the values of $\bar{SDD}$ , ${\bar{M}}_{1}$ , $\bar{A}$ and ultimately these codescriptors gives CE for any number of Li atoms with the help of equations 4 to 6.

CoM polynomial of 2D porous graphene

In this section, we develop analytical expressions for topological codescriptors (TCD) derived from the CoM Polynomial, for various porous graphene structures. Subsequently, we numerically tabulate TCDs for these structures, followed by graphical representation and analysis to identify trends across different structures. A porous graphene is a material with nanopores in the plane, and the pore size and distribution of the porous graphene are determined by the method of production. Bieri et al.²⁵ successfully transformed graphene structure into a porous graphene structure by strategically removing phenyl rings at regular intervals. The resulting structural relationship between this network (represented by bold lines) and the original graphene (depicted with thin lines) is visually illustrated in Figure 6.

Figure 6.

Structure of porous graphene.

The unique properties of porous graphene, applicable across diverse fields, have recently sparked a surge in scientific interest. This has driven researchers to develop efficient methods for synthesizing well-defined porous graphene with controlled pore structures. By adopting various approaches, researchers have succeeded in generating diverse types of porous graphene structures such as ordered porous graphene nanoribbon⁵³ and trigonal porous graphene⁵⁴ as evidenced by the literature.⁵⁵ These structures inspired us to conduct a topological study using CoM Polynomial techniques. The structures examined in this section are depicted in Figure 7.

Figure 7.

Structure of linear and trigonal porous graphene.

CoM polynomial of linear porous graphene (LPG)

As a first step, we determine the expression for the CoM polynomial for LPG.

Theorem 1: The CoM Polynomial for LPG given by

\begin{matrix} CoM (LPG; x, y) = \\ = (98 p^{2} + 129 p + 37) x^{2} y^{2} \\ + (140 p^{2} + 108 p + 16) x^{2} y^{3} + (50 p^{2} + 10 p) x^{3} y^{3} \end{matrix}

Proof: It is straightforward to deduce the following from Figure 8:

\begin{matrix} E_{22} = {uv \in E (LPG) | d (u) = 2, d (v) = 2}, \\ E_{23} = {uv \in E (LPG) | d (u) = 2, d (v) = 3}, \\ E_{33} = {uv \in E (LPG) | d (u) = 3, d (v) = 3} such that \end{matrix}

| E_{22} | = 4 (p + 2), | E_{23} | = 4 (5 p + 1), | E_{33} | = 5 p + 1

Figure 8.

Molecular graphs G $(LPG)$ of linear porous graphene.

In a similar way, LPG’s vertex set can be divided into two classes according to their degrees.

\begin{matrix} n_{2} = | V_{2} | = 14 p + 10, n_{3} = | V_{3} | = 10 p + 2, \\ n_{3} = | V_{3} | = 10 p + 2 \end{matrix}

Using (1), we have

\begin{matrix} {\bar{m}}_{23} = (14 p + 10) (10 p + 2) - (20 p + 4) \\ = 140 p^{2} + 108 p + 16 \end{matrix}

\begin{matrix} {\bar{m}}_{22} = \frac{(14 p + 10) (14 p + 9)}{2} - 4 (p + 2) \\ = 98 p^{2} + 129 p + 37 \end{matrix}

Similarly, ${\bar{m}}_{33} = | {\bar{E}}_{33} | = = 50 p^{2} + 10 p$

By definition of Co-M-polynomial

CoM (G; x, y) = \sum_{i \leq j} {\bar{m}}_{ij} (G) x^{i} y^{j}

\begin{matrix} CoM (LPG; x, y) = \sum_{2 \leq 2} {\bar{m}}_{22} x^{2} y^{2} \\ + \sum_{2 \leq 3} {\bar{m}}_{23} x^{2} y^{3} + \sum_{3 \leq 3} {\bar{m}}_{33} x^{3} y^{3} \\ = | {\bar{E}}_{22} | x^{2} y^{2} + | {\bar{E}}_{23} | x^{2} y^{3} + | {\bar{E}}_{33} | x^{3} y^{3} \\ = (98 p^{2} + 129 p + 37) x^{2} y^{2} \\ + (140 p^{2} + 108 p + 16) x^{2} y^{3} \\ + (50 p^{2} + 10 p) x^{3} y^{3} \end{matrix}

Thus, the result.

The next proposition uses Theorem 1 to recover some degree-based TCD of the LPG.

Proposition 1: The topological codescriptors for LPG are given by

$1 . {\bar{M}}_{1} (LPG) = 1392 p^{2} + 1116 p + 228$ .

$2 . {\bar{M}}_{2} (LPG) = 1682 p^{2} + 1254 p + 244$ .

$3 . m {\bar{M}}_{2} (LPG) = \frac{961}{18} p^{2} + \frac{1849}{36} p + \frac{143}{12}$

$4 . \bar{ReZ G_{3}} (LPG) = 8468 p^{2} + 5844 p + 1072$ .

$5 . \bar{F} (LPG) = 1402 p^{2} + 1038 p + 212$

$6 . {\bar{R}}_{k} (LPG) = 2^{2 k} (98 p^{2} + 129 p + 37) + 2^{k} 3^{k} (140 p^{2} + 108 p + 16) + 3^{2 k} (50 p^{2} + 10 p)$ .

$7 . {\bar{RR}}_{k} (LPG) = (98 p^{2} + 129 p + 37) \frac{1}{{(2)}^{2 k}} + (140 p^{2} + 108 p + 16) \frac{1}{{(2)}^{k} {(3)}^{k}} + (50 p^{2} + 10 p) \frac{1}{{(3)}^{2 k}}$

$8 . \bar{SDD} (LPG) = \frac{1798}{3} p^{2} + 512 p + \frac{326}{3}$

$9 . \bar{H} (LPG) = \frac{365}{3} p^{2} + \frac{3331}{30} p + \frac{249}{10}$

$10 . \bar{I} (LPG) = 341 p^{2} + \frac{1368}{5} p + \frac{281}{5}$

$11 . \bar{A} (LPG) = \frac{79, 153}{32} p^{2} + \frac{64, 317}{32} p + 424$

Proof: Let

\begin{matrix} f (x, y) = CoM (LPG; x, y) \\ = (98 p^{2} + 129 p + 37) x^{2} y^{2} \\ + (140 p^{2} + 108 p + 16) x^{2} y^{3} \\ + (50 p^{2} + 10 p) x^{3} y^{3}, \end{matrix}

then

$D_{x} f (x, y) = 2 (98 p^{2} + 129 p + 37) x^{2} y^{2} + 2 (140 p^{2} + 108 p + 16) x^{2} y^{3} + 3 (50 p^{2} + 10 p) x^{3} y^{3}$ .

$D_{y} f (x, y) = 2 (98 p^{2} + 129 p + 37) x^{2} y^{2} + 3 (140 p^{2} + 108 p + 16) x^{2} y^{3} + 3 (50 p^{2} + 10 p) x^{3} y^{3}$ .

$(D_{x} + D_{y}) (f (x, y)) = 4 (98 p^{2} + 129 p + 37) x^{2} y^{2} + 5 (140 p^{2} + 108 p + 16) x^{2} y^{3} + 6 (50 p^{2} + 10 p) x^{3} y^{3}$ .

$D_{y} D_{x} (f (x, y)) = 4 (98 p^{2} + 129 p + 37) x^{2} y^{2} + 6 (140 p^{2} + 108 p + 16) x^{2} y^{3} + 9 (50 p^{2} + 10 p) x^{3} y^{3}$ .

$(D_{x}^{2} + D_{y}^{2}) (f (x, y)) = 4 (98 p^{2} + 129 p + 37) x^{2} y^{2} + 4 (140 p^{2} + 108 p + 16) x^{2} y^{3} + 9 (50 p^{2} + 10 p) x^{3} y^{3}$ .

${D_{x}}^{k} {D_{y}}^{k} (f (x, y)) = 2^{2 k} (98 p^{2} + 129 p + 37) x^{2} y^{2} + 2^{k} 3^{k} (140 p^{2} + 108 p + 16) x^{2} y^{3} + 3^{2 k} (50 p^{2} + 10 p) x^{3} y^{3}$ .

$D_{x} D_{y} (D_{x} + D_{y}) (f (x, y) = 16 (98 p^{2} + 129 p + 37) x^{2} y^{2} + 30 (140 p^{2} + 108 p + 16) x^{2} y^{3} + 54 (50 p^{2} + 10 p) x^{3} y^{3}$ .

$S_{x} S_{y} (f (x, y)) = (98 p^{2} + 129 p + 37) \frac{1}{(2) (2)} x^{2} y^{2} + (140 p^{2} + 108 p + 16) \frac{1}{(2) (3)} x^{2} y^{3} + (50 p^{2} + 10 p) \frac{1}{(3) (3)} x^{3} y^{3}$ .

${S_{x}}^{k} {S_{y}}^{k} (f (x, y)) = (98 p^{2} + 129 p + 37) \frac{1}{{(2)}^{2 k}} x^{2} y^{2} + (140 p^{2} + 108 p + 16) \frac{1}{{(2)}^{k} {(3)}^{k}} x^{2} y^{3} + (50 p^{2} + 10 p) \frac{1}{{(3)}^{2 k}} x^{3} y^{3}$ .

$(S_{y} D_{x} + S_{x} D_{y}) (f (x, y)) = 2 (98 p^{2} + 129 p + 37) x^{2} y^{2} + \frac{13}{6} (140 p^{2} + 108 p + 16) x^{2} y^{3} + 2 (50 p^{2} + 10 p) x^{3} y^{3}$ .

$S_{x} J (f (x, y)) = (98 p^{2} + 129 p + 37) \frac{1}{4} x^{4} + (140 p^{2} + 108 p + 16) \frac{1}{5} x^{5} + (50 p^{2} + 10 p) \frac{1}{6} x^{6}$ .

$S_{x} J D_{y} D_{x} (f (x, y)) = (98 p^{2} + 129 p + 37) x^{4} + (140 p^{2} + 108 p + 16) (\frac{6}{5}) x^{5} + (50 p^{2} + 10 p) (\frac{3}{2}) x^{6}$ .

${S_{x}}^{3} Q_{- 2} J {D_{x}}^{3} {D_{y}}^{3} (f (x, y)) = 8 (98 p^{2} + 129 p + 37) x^{2} + 8 (140 p^{2} + 108 p + 16) x^{3} + \frac{729}{64} (50 p^{2} + 10 p) x^{4}$ .

With the help of Table 1 we get the following and Table 5,

${\bar{M}}_{1} (LPG) = {(D_{x} + D_{y}) f (x, y) |}_{x = y = 1} = 1392 p^{2} + 1116 p + 228$ .

${\bar{M}}_{2} (LPG) = {D_{x} D_{y} f (x, y) |}_{x = y = 1} = 1682 p^{2} + 1254 p + 244$ .

$m {\bar{M}}_{2} (LPG) = {S_{x} S_{y} f (x, y) |}_{x = y = 1} = \frac{961}{18} p^{2} + \frac{1849}{36} p + \frac{143}{12}$

$\bar{ReZ G_{3}} (LPG) = {D_{x} D_{y} (D_{x} + D_{y}) f (x, y) |}_{x = y = 1} = 8468 p^{2} + 5844 p + 1072$ .

$\bar{F} (LPG) = {(D_{x}^{2} + D_{y}^{2}) (f (x, y)) |}_{x = y = 1} = 1402 p^{2} + 1038 p + 212$ .

${\bar{R}}_{k} (LPG) = {{D_{x}}^{k} {D_{y}}^{k} f (x, y) |}_{x = y = 1} = 2^{2 k} (98 p^{2} + 129 p + 37) + 2^{k} 3^{k} (140 p^{2} + 108 p + 16) + 3^{2 k} (50 p^{2} + 10 p)$ .

${\bar{RR}}_{k} (LPG) = {{S_{x}}^{k} {S_{y}}^{k} f (x, y) |}_{x = y = 1} = (98 p^{2} + 129 p + 37) \frac{1}{{(2)}^{2 k}} + (140 p^{2} + 108 p + 16) \frac{1}{{(2)}^{k} {(3)}^{k}} + (50 p^{2} + 10 p) \frac{1}{{(3)}^{2 k}}$ .

$\bar{SDD} (LPG) = {(S_{y} D_{x} + S_{x} D_{y}) f (x, y) |}_{x = y = 1} = \frac{1798}{3} p^{2} + 512 p + \frac{326}{3}$

$\bar{H} (LPG) = {2 S_{x} Jf (x, y) |}_{x = y = 1} = \frac{365}{3} p^{2} + \frac{3331}{30} p + \frac{249}{10}$

$\bar{I} (LPG) = {S_{x} J D_{x} D_{y} f (x, y) |}_{x = y = 1} = 341 p^{2} + \frac{1368}{5} p + \frac{281}{5}$

$\bar{A} (LPG) = {{S_{x}}^{3} Q_{- 2} J {D_{x}}^{3} {D_{y}}^{3} f (x, y) |}_{x = y = 1} = \frac{79, 153}{32} p^{2} + \frac{64, 317}{32} p + 424$ .

Table 5.

Numerical values of TCD of linear porous graphene for $p = 1 - 10$ .

$(p)$	${\bar{M}}_{1} (LPG)$	${\bar{M}}_{2} (LPG)$	$m {\bar{M}}_{2} (LPG)$	$\bar{ReZ G_{3}} (LPG)$	$\bar{F} (LPG)$	${\bar{R}}_{- \frac{1}{2}} (LPG)$	$\bar{SDD}$ (LPG)	$\bar{H} (LPG)$	$\bar{I} (LPG)$	$\bar{A} (LPG)$
$(1)$	2736	3180	116.67	15,384	2652	259.78	1220	128.80	670.80	4907.44
$(2)$	8028	9480	328.19	46,632	7896	740.16	3530	366.82	1967.40	14,337.94
$(3)$	16,104	19,144	646.50	94,816	15,944	1466.2	7038.67	726.50	3946	28,715.50
$(4)$	26,964	32,172	1071.58	159,936	26,796	2437.9	11,746	1207.85	6606.60	48,040.12
$(5)$	40,608	48,564	1603.44	241,992	40,452	3655.2	17,652	1810.87	9949.20	72,311.81
$(6)$	57,036	68,320	2242.08	340,984	56,912	5118.2	24,756.67	2535.55	13,973.80	101,530.56
$(7)$	76,248	91,440	2987.50	456,912	76,176	6826.8	33,060	3381.90	18,680.40	135,696.38
$(8)$	98,244	117,924	3839.69	589,776	98,244	8781.0	42,562	4349.92	24,069	174,809.25
$(9)$	123,024	147,772	4798.67	739,576	123,116	10,981.88	53,262.67	5439.6	30,139.60	218,869.19
$(10)$	150,588	180,984	5864.42	906,312	150,792	13,426.42	65,162	6650.95	36,892.20	267,876.19

CoM polynomial of triangular porous graphene (TPG)

Theorem 2: The CoM Polynomial for TPG given by

\begin{matrix} CoM (TPG; x, y) = \\ = (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} \\ + (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} \\ + (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3} \end{matrix}

Proof: According to Figure 9,

$| E (TPG) | = 30 p^{2} + 27 p - 15, | V (TPG) | = 24 p^{2} + 24 p - 12$ . The $| E (TPG) |$ classified into 3 classes due to type of the vertex degrees:

\begin{matrix} E_{22} = {uv \in E (TPG) | d (u) = 2, d (v) = 2}, \\ E_{23} = {uv \in E (TPG) | d (u) = 2, d (v) = 3}, \\ E_{33} = {uv \in E (TPG) | d (u) = 3, d (v) = 3} \end{matrix}

Figure 9.

Molecular graphs G $(TPG)$ of triangular porous graphene.

such tha

\begin{matrix} | E_{22} | = 12 p, | E_{23} | = 24 p^{2} + 12 p - 12, \\ | E_{33} | = 6 p^{2} + 3 p - 3 \end{matrix}

and, TPG’s vertex set classified into two classes as,

n_{2} = | V_{2} | = 12 p^{2} + 18 p - 6, n_{3} = | V_{3} | = 12 p^{2} + 6 p - 6

Using (1), we have

\begin{matrix} {\bar{m}}_{23} = (12 p^{2} + 18 p - 6) (12 p^{2} + 6 p - 6) \\ - (24 p^{2} + 12 p - 12) \\ = 144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48 \end{matrix}

\begin{matrix} {\bar{m}}_{22} = \frac{(12 p^{2} + 18 p - 6) (12 p^{2} + 18 p - 7)}{2} - (12 p) \\ = 72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21 \end{matrix}

Similarly, ${\bar{m}}_{33} = | {\bar{E}}_{33} | = = 72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24$

By definition of CoM Polynomial

CoM (G; x, y) = \sum_{i \leq j} {\bar{m}}_{ij} (G) x^{i} y^{j}

\begin{matrix} CoM (TPG; x, y) = \sum_{2 \leq 2} {\bar{m}}_{22} x^{2} y^{2} \\ + \sum_{2 \leq 3} {\bar{m}}_{23} x^{2} y^{3} + \sum_{3 \leq 3} {\bar{m}}_{33} x^{3} y^{3} \\ = | {\bar{E}}_{22} | x^{2} y^{2} + | {\bar{E}}_{23} | x^{2} y^{3} + | {\bar{E}}_{33} | x^{3} y^{3} \\ = (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} \\ + (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} \\ + (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3} . \end{matrix}

Proposition 2: The topological codescriptors for TPG are given by

1. ${\bar{M}}_{1} (TPG) = 1392 p^{2} + 1116 p + 228$ .

2. ${\bar{M}}_{2} (TPG) = 1682 p^{2} + 1254 p + 244$ .

3. $m {\bar{M}}_{2} (TPG) = \frac{961}{18} p^{2} + \frac{1849}{36} p + \frac{143}{12}$

4. $\bar{ReZ G_{3}} (TPG) = 8468 p^{2} + 5844 p + 1072$ .

5. $\bar{F} (TPG) = 3504 p^{2} + 2616 p + 504$

6. ${\bar{R}}_{k} (TPG) = 2^{2 k} (98 p^{2} + 129 p + 37) + 2^{k} 3^{k} (140 p^{2} + 108 p + 16) + 3^{2 k} (50 p^{2} + 10 p)$ .

7. ${\bar{RR}}_{k} (TPG) = (98 p^{2} + 129 p + 37) \frac{1}{{(2)}^{2 k}} + (140 p^{2} + 108 p + 16) \frac{1}{{(2)}^{k} {(3)}^{k}} + (50 p^{2} + 10 p) \frac{1}{{(3)}^{2 k}}$ .

8. $\bar{SDD} (TPG) = \frac{1798}{3} p^{2} + 512 p + \frac{326}{3}$

9. $\bar{H} (TPG) = \frac{365}{3} p^{2} + \frac{3331}{30} p + \frac{249}{10}$

10. $\bar{I} (TPG) = 341 p^{2} + \frac{1368}{5} p + \frac{281}{5}$

11. $\bar{A} (TPG) = \frac{79, 153}{32} p^{2} + \frac{64, 317}{32} p + 424$

Proof: Let

\begin{matrix} f (x, y) = CoM (TPG; x, y) \\ = (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} \\ + (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} \\ + (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3}, \end{matrix}

then

$D_{x} f (x, y) = 2 (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} + 2 (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} + 3 (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3}$ .

$D_{y} f (x, y) = 2 (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} + 3 (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} + 3 (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3}$ .

$(D_{x} + D_{y}) (f (x, y)) = 4 (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} + 5 (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} + 6 (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3}$ .

$D_{y} D_{x} (f (x, y)) = 4 (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} + 6 (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} + 9 (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3}$ .

$(D_{x}^{2} + D_{y}^{2}) (f (x, y)) = 8 (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} + 13 (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} + 18 (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3}$ .

${D_{x}}^{k} {D_{y}}^{k} (f (x, y)) = 2^{2 k} (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} + 2^{k} 3^{k} (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} + 3^{2 k} (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3}$ .

$D_{x} D_{y} (D_{x} + D_{y}) (f (x, y) = 16 (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} + 30 (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} + 54 (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3}$ .

$S_{x} S_{y} (f (x, y)) = (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) \frac{1}{(2) (2)} x^{2} y^{2} + (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) \frac{1}{(2) (3)} x^{2} y^{3} + (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) \frac{1}{(3) (3)} x^{3} y^{3}$ .

${S_{x}}^{k} {S_{y}}^{k} (f (x, y)) = (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) \frac{1}{{(2)}^{2 k}} x^{2} y^{2} + (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) \frac{1}{{(2)}^{k} {(3)}^{k}} x^{2} y^{3} + (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) \frac{1}{{(3)}^{2 k}} x^{3} y^{3}$ .

$(S_{y} D_{x} + S_{x} D_{y}) (f (x, y)) = 2 (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} y^{2} + \frac{13}{6} (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{2} y^{3} + 2 (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{3} y^{3}$ .

$S_{x} J (f (x, y)) = (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) \frac{1}{4} x^{4} + (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) \frac{1}{5} x^{5} + (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) \frac{1}{6} x^{6}$ .

$S_{x} J D_{y} D_{x} (f (x, y)) = (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{4} + (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) (\frac{6}{5}) x^{5} + (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) (\frac{3}{2}) x^{6}$ .

${S_{x}}^{3} Q_{- 2} J {D_{x}}^{3} {D_{y}}^{3} (f (x, y)) = 8 (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) x^{2} + 8 (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) x^{3} + \frac{729}{64} (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) x^{4}$ .

Based on Table 1, we can find (Table 6)

${\bar{M}}_{1} (TPG) = {(D_{x} + D_{y}) f (x, y) |}_{x = y = 1} = 1440 p^{4} + 2736 p^{3} - 360 p^{2} - 1548 p + 468$ .

${\bar{M}}_{2} (TPG) = {D_{x} D_{y} f (x, y) |}_{x = y = 1} = 1800 p^{4} + 3240 p^{3} - 618 p^{2} - 1830 p + 588$ .

$m {\bar{M}}_{2} (TPG) = {S_{x} S_{y} f (x, y) |}_{x = y = 1} = 50 p^{4} + 110 p^{3} + \frac{11}{3} p^{2} - \frac{755}{12} p + \frac{191}{12}$

$\bar{ReZ G_{3}} (TPG) = {D_{x} D_{y} (D_{x} + D_{y}) f (x, y) |}_{x = y = 1} = 9360 p^{4} + 15, 984 p^{3} - 4020 p^{2} - 9012 p + 3072$ .

$\bar{F} (T P G) = (D_{x}^{2} + D_{y}^{2}) (f (x, y)) |_{x = y = 1} = 3744 p^{4} + 6768 p^{3} - 1296 p^{2} - 3816 p + 1224$ .

${\bar{R}}_{k} (TPG) = {{D_{x}}^{k} {D_{y}}^{k} f (x, y) |}_{x = y = 1} = 2^{2 k} (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) + 2^{k} 3^{k} (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) + 3^{2 k} (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24)$ .

${\bar{RR}}_{k} (TPG) = {{S_{x}}^{k} {S_{y}}^{k} f (x, y) |}_{x = y = 1} = (72 p^{4} + 216 p^{3} + 84 p^{2} - 129 p + 21) \frac{1}{{(2)}^{2 k}} + (144 p^{4} + 288 p^{3} - 60 p^{2} - 156 p + 48) \frac{1}{{(2)}^{k} {(3)}^{k}} + (72 p^{4} + 72 p^{3} - 66 p^{2} - 42 p + 24) \frac{1}{{(3)}^{2 k}}$ .

$\bar{SDD} (TPG) = {(S_{y} D_{x} + S_{x} D_{y}) f (x, y) |}_{x = y = 1} = 600 p^{4} + 1200 p^{3} - 94 p^{2} - 680 p + 194$

$\bar{H} (TPG) = {2 S_{x} Jf (x, y) |}_{x = y = 1} = \frac{588}{5} p^{4} + \frac{1236}{5} p^{3} - 4 p^{2} - \frac{1409}{10} p + \frac{377}{10}$

$\bar{I} (TPG) = {S_{x} J D_{x} D_{y} f (x, y) |}_{x = y = 1} = \frac{1764}{5} p^{4} + \frac{3348}{5} p^{3} - 87 p^{2} - \frac{1896}{5} p + \frac{573}{5}$

$\bar{A} (TPG) = {{S_{x}}^{3} Q_{- 2} J {D_{x}}^{3} {D_{y}}^{3} f (x, y) |}_{x = y = 1} = \frac{20, 385}{8} p^{4} + \frac{38, 817}{8} p^{3} - \frac{17, 913}{32} p^{2} - \frac{88, 269}{32} p + \frac{6603}{8}$ .

Table 6.

Numerical values of TCD of triangular porous graphene for $p = 1 - 10$ .

$(p)$	${\bar{M}}_{1} (TPG)$	${\bar{M}}_{2} (TPG)$	$m {\bar{M}}_{2} (TPG)$	$\bar{ReZ G_{3}} (TPG)$	$\bar{F} (TPG)$	${\bar{R}}_{- \frac{1}{2}} (TPG)$	$\bar{SDD}$ (TPG)	$\bar{H} (TPG)$	$\bar{I} (TPG)$	$\bar{A} (TPG)$
$(1)$	2736	3180	116.67	15,384	6624	259.78	1220	257.60	670.8	4907.44
$(2)$	40,860	49,176	1584.75	246,600	102,456	3633.0	17,658	3599.10	10,009.8	72,656.44
$(3)$	183,096	222,816	6880.17	1,129,584	464,112	15,932.0	78,308	15,779.0	44,850.0	324,917.62
$(4)$	532,260	651,540	19,662.92	3,321,840	135,684	45,780.0	226,370	45,336.50	130,377.0	943,691.25
$(5)$	1,225,728	1,505,988	44,793.00	7,705,512	313,574	104,650.0	519,444	103,633.2	300,243.6	2,172,132.56
$(6)$	2,435,436	3,000,000	88,330.42	15,387,384	624,578	206,880.0	1,029,530	204,853.1	596,569.8	4,314,551.81
$(7)$	4,367,880	5,390,616	157,535.17	27,698,880	112,217	369,630.0	1,843,028	366,002.6	1,069,942.8	7,736,414.25
$(8)$	7,264,116	8,978,076	260,867.25	46,196,064	186,883	612,950.0	3,060,738	606,910.5	1,779,417.0	12,864,340.12
$(9)$	11,399,760	14,105,820	407,986.67	72,659,640	293,601	959,700.0	4,797,860	950,228.0	2,792,514.0	20,186,104.69
$(10)$	17,084,988	21,160,488	609,753.42	109,094,952	440,414	1,435,619.	7,183,994	1,421,428.	4,185,222.6	30,250,638.19

Discussion

The calculation methods used in this study, namely the CoM Polynomial computations and curvilinear regression analysis, should be compared with other methods in the literature to benchmark their performance and highlight their advantages and limitations.

One method for comparison is density functional theory (DFT) calculations, which have been widely used to study lithium clusters. While DFT calculations provide accurate results, they are computationally expensive and limited to small cluster sizes. In contrast, the CoM Polynomial computations offer a more efficient and scalable approach for larger clusters.

Another method for comparison is machine learning-based approaches, such as neural networks and Gaussian process regression. These methods have been applied to predict cluster energies and properties, but require large amounts of training data and can be prone to overfitting. The curvilinear regression analysis used in this study offers a more interpretable and robust approach, especially when combined with the CoM Polynomial computations.

Additionally, other computational methods such as Monte Carlo simulations and molecular dynamics simulations can also be used to study lithium clusters. These methods can provide insights into the dynamic behavior and thermodynamic properties of the clusters, but are often limited by their computational cost and scalability.

A detailed comparison of these methods would require a thorough analysis of their accuracy, computational cost, and applicability to different cluster sizes and environments. This would provide a comprehensive understanding of the strengths and limitations of each method and highlight the advantages of the CoM Polynomial computations and curvilinear regression analysis used in this study.

By comparing the calculation methods used in this study with other methods in the literature, we can gain a deeper understanding of the complex behavior of lithium clusters and develop more accurate and efficient computational tools for predicting their properties and behavior.

Conclusions

Designing materials whose desired properties lead to the discovery of new structures has become increasingly difficult without theoretical guidance. Identifying key descriptors describing the structure of a material with the target property is the first step in this process. Therefore in this article, by utilizing codescriptors derived from CoM Polynomial and employing curvilinear regression analysis, we have theoretically predicted cluster energy of Lithium Cluster and we can summarize the results obtained in this article as follows:

Among ten codescriptors, Symmetric Division $(\bar{SDD})$ with $R^{2} = 0.963$ emerges as the most effective for predicting the Cluster Energy of $L i_{n} (n = 4 - 10)$ clusters within a linear regression model. For quadratic regression, First Zagreb $({\bar{M}}_{1})$ with $R^{2} = 0.982$ takes the crown, while Augmented Zagreb $(\bar{A}$ ) with $R^{2} = 0.995$ reigns supreme in the cubic regression model.

For the studied property of $L i_{n} (n = 4 - 10)$ clusters, highly significant regression equations (4) to (6) have been developed and their predictive value has been demonstrated statistically (Table 4).

We derived generalized analytical expressions for degree-based codescriptors using the CoM Polynomial method applied to Linear and Triangular Porous Graphene structures. Additionally, we present graphical comparisons of these codescriptors for each structure, allowing for insightful visualization and analysis (Figures 10 and 11). Future research directions are illustrated in Figure 12.

Figure 10.

Graphical visualization of Table 5.

Figure 11.

Graphical visualization of Table 6.

Figure 12.

Future work.

Footnotes

Handling editor: Sharmili Pandian

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The author(s) gratefully acknowledge Qassim University, represented by the “Deanship of Scientific Research, on the financial support for this research under the number (ENUC-2022-1-2-J-29899) during the academic year 1444 AH/2022 AD.”

ORCID iD

Parvez Ali

Data availability statement

In this paper, the authors provided all raw data to support their conclusions.

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Prediction of cluster energy of lithium clusters from codescriptors with application in 2D porous graphene

Abstract

Keywords

Introduction

Preliminaries

Relationship between the cluster energy with various topological codescriptor of lithium clusters

CoM polynomial of 2D porous graphene

CoM polynomial of linear porous graphene (LPG)

CoM polynomial of triangular porous graphene (TPG)

Discussion

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Data availability statement

References