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Article

Neighbourhood Sum Degree-Based Indices and Entropy Measures for Certain Family of Graphene Molecules

by
Jun Yang
1,
Julietraja Konsalraj
2,* and
Arul Amirtha Raja S.
2
1
School of Economics and Law, Chaohu University, Chaohu 238000, China
2
Department of Mathematics, St. Joseph’s College of Engineering, OMR, Chennai 600119, India
*
Author to whom correspondence should be addressed.
Molecules 2023, 28(1), 168; https://doi.org/10.3390/molecules28010168
Submission received: 2 December 2022 / Revised: 20 December 2022 / Accepted: 21 December 2022 / Published: 25 December 2022
(This article belongs to the Special Issue Study of Molecules in the Light of Spectral Graph Theory)

Abstract

:
A topological index (TI) is a real number that defines the relationship between a chemical structure and its properties and remains invariant under graph isomorphism. TIs defined for chemical structures are capable of predicting physical properties, chemical reactivity and biological activity. Several kinds of TIs have been defined and studied for different molecular structures. Graphene is the thinnest material known to man and is also extremely strong while being a good conductor of heat and electricity. With such unique features, graphene and its derivatives have found commercial uses and have also fascinated theoretical chemists. In this article, the neighbourhood sum degree-based M-polynomial and entropy measures have been computed for graphene, graphyne and graphdiyne structures. The proper analytical expressions for these indices are derived. The obtained results will enable theoretical chemists to study these exciting structures further from a structural perspective.

1. Introduction

The study of the association between the structural formula of chemical compounds and their corresponding topological graphs was a ground-breaking advancement in investigating chemical structures. Since then, the basic concepts of chemical graph theory have been developed as a combination of principles from both mathematics and chemistry. Chemical graph theory enables mathematicians and chemists to theoretically analyse molecular compounds by depicting their structures as graphs. The vertices represent the atoms of molecules or molecule collections in a chemical graph. The edges of a chemical graph indicate the relationships between the chemical objects and generally represent reactions, reaction mechanisms, chemical bonds, or other transformations in chemical entities.
Graph theoretical tools have been gaining popularity as the primary techniques for the theoretical study of chemical compounds. QSAR/QSPR techniques, in particular, are regarded as useful computational and quasi-strategies for trying to predict the characteristics of chemical substances. These techniques are crucial in the development of new and more effective herbicides because their properties can be estimated prior to synthesising and thus influence the design. Furthermore, experimental measurements can be replaced by QSPR/QSAR models, which are less expensive and time-consuming [1]. In this context, Topological indices provide a quantitative characterisation of molecular topology.
A topological index (TI) is a real number that defines the relationship between a chemical structure and its properties and remains invariant under graph isomorphism. TIs defined for chemical structures are capable of predicting physical properties, chemical reactivity, and biological activity. Several kinds of TIs have been defined and studied for different molecular structures. Graphene is the thinnest material known to man and is also extremely strong while being a good conductor of heat and electricity. With such unique features, graphene and its derivatives have found commercial uses and have also fascinated theoretical chemists.
The first occurrence of a topological index in the literature is in the pioneering work of the eminent chemist Wiener. He described the boiling point of alkanes in terms of a path number W, which is the sum of the distances between any two atoms within the molecule. This path number later became known as the Wiener index and has been studied extensively [2]. This ground-breaking work kickstarted the research on topological indices, and its mathematical investigation began in the 1970s. Since the behaviour and properties of molecules rely heavily on the corresponding molecular structures, TIs have now been established as major molecular indices in theoretical chemistry.
A considerable number of topological indices have been proven to display a strong correlation with several properties of chemical compounds. Due to their ability to characterise a large variety of physical and chemical properties, the study of topological indices has wide-ranging applications in various fields, including computer-assisted drug discovery, deriving multi-linear regression models [3], aromatic sextet theory [4], and thermochemistry [5]. The comparative ease of using molecular TIs to determine the physicochemical properties as opposed to the complex quantum chemical calculations has also found several applications for these TIs [6]. Thus, it becomes vital to determine the various molecular indices of molecules so that suitable indices are applied to attain the desired correlations between their properties and structures. There are several kinds of TIs that are currently defined and studied in the literature. The most commonly used TIs among them are distance and degree-based indices. A topological index is said to be distance-based if its computation involves distance between vertices. A degree-based topological index is a sub-class of TIs where the index is computed based on the degrees of the end vertices of the molecular graph.
One of the most efficient and convenient ways to compute TIs is by using algebraic polynomials, where the estimations of multiple TIs are reduced to the computation of a single polynomial. For example, the Hosoya polynomial [7] is widely applied for calculating distance-based TIs. Deutsch and Klavžar [8] created a similar polynomial for computing degree-based indices. In this technique, a single polynomial, referred to as the M-polynomial, is used to compute several prominent degree-based indices. A significant number of articles in the literature deal with M-polynomial and the computation of degree-based indices using it. The degree-based entropy measures are computed for these structures [9].
The neighbourhood sum degree-based TIs are one of the recent developments in analysing chemical compounds using graph theory. They have been used for QSPR and statistics regression analysis for predicting specific chemical properties of molecular graphs [10,11]. The degree-based entropy measures studied and widely applied in graph theory are considered information functionals in the study of networks. The applications of entropy network measures range from the description of the structure of a molecule quantitatively to the exploration of biological and chemical features of molecular graphs. Presently, very few articles in the literature demonstrate the computation of neighbourhood sum degree-based M-Polynomial and neighbourhood sum degree-based entropy measures. Hence, this article is primarily concerned with computing the neighbourhood sum degree-based M-Polynomial and neighbourhood sum degree-based entropy measures for three prominent structures: graphene, graphyne and graphdiyne. This computation has not been conducted for these structures yet. Therefore, the computed results will provide a unique viewpoint for researchers to study these structures. The findings can also be extended to other large structures with a similar structural base.

2. Results

2.1. Mathematical Concepts

Throughout this research, we consider only a simple and connected graph without multiple edges and self-loops. The graph Γ is said to be a connected graph with vertex set V ( Γ ) and edge set E ( Γ ) . The degree of the vertex is represented as d v .

2.2. Neighbourhood Degree Sum-Based Indices

The neighbourhood sum degree-based TIs are denoted by N Γ ( v ) . The neighbourhood sum degree of the molecular graph is represented as | N Γ ( v ) | = d v . The N Γ ( v ) denotes the sum of the degrees of the neighbouring vertices of v . Let us define the neighbourhood sum degree-based M -polynomial of Γ ,
N M ( Γ ) = 𝒾 𝒿 ( N u m b e r   o f   a l l   e d g e s   u v   s u c h   t h a t   u = 𝒾 ,   v = 𝒿 ) 𝓇 i 𝓉 j .
The degree-based TIs and neighbourhood degree sum-based (ND) TIs are depicted as [10,11]
D ( Γ ) = u v ϵ E ( Γ ) g ( φ u φ v )
and
N M ( Γ ) = u v ϵ E ( Γ ) g ( ω u ω v )
The derivations of N M -Polynomial are listed below in Table 1.

3. Neighbourhood Degree Sum-Based Entropy Measures

In his seminal work, Shannon defined entropy as a measure of the unpredictable nature of relevant information or a way of measuring a system’s uncertainty. This paper laid the foundation for modern information theory. The entropy formulae have been used to quantify a network’s structural informativeness [12]. Though information theory was initially used exclusively in electrical engineering and linguistics, its versatile nature found applications in life sciences like chemistry and biology [13] and in graph theory for chemical networks. The notion of graph entropy was proposed to quantify the topological information of chemical networks and graphs.
Rashevsky [14] developed the concept of graph entropy depending on the vertex orbits. The graph entropy measures enable mathematicians to relate graph components such as edges and vertices with probability distributions, categorised as intrinsic and extrinsic measures. Graph entropies have wide-ranging applications in many fields, including chemistry, ecology, sociology, and biology [15,16]. Dehmer introduced graph entropies that captured structural information based on information functionals and studied their properties [17,18]. Estrada et al. [19] introduced a physically-sound graph entropy measure and analysed the walk-based graph entropies [20]. The applications of entropy network measures range from quantitatively describing a molecular structure to exploring biological and chemical features of molecular graphs. Entropy measures have several applications in the field of chemical graph theory. They are used to analyse complex networks and their chemical properties.
Let Γ be a graph with vertex v i and d i be the degree of v i for the given edge u i v j , then one can define
P i j = w u i v j j = 1 d i w u i v j
where w ( u i v j ) be the weight of the edge u i v j and w ( u i v j ) > 0 . The node entropy is defined as
ENT Γ v i = j = 1 d i P i j log P i j
For an edge-weighted graph Γ = ( V , E , w ) , the entropy measure of Γ is defined as [21]
ENT Γ ( Γ , w ) = u v E ( Γ ) P u v log P u v
where
P u v = w ( u v ) u v E ( Γ ) w ( u v )
ENT χ ( Γ ) = u v E ( Γ ) P u , v log P u , v
= u v E ( Γ ) F d u , d v χ ( Γ ) log F d u , d v χ ( Γ )
= 1 χ ( Γ ) u v E ( Γ ) F d u , d v log F d u , d v χ ( Γ )
= 1 χ ( Γ ) u v E ( Γ ) F d u , d v log F d u , d v ( log χ ( Γ ) )
= log χ ( Γ ) 1 χ ( Γ ) u v E ( Γ ) F d u , d v log F d u , d v
where T I ( Γ ) = χ ( Γ ) .

4. Computing the Neighbourhood Sum Degree-Based M-polynomial for β-Graphene

In this section, the proper analytical expressions of neighbourhood degree sum-based indices and entropy measures are computed using the M-polynomial for β-Graphene. Graphene is a carbon allotrope, a two-dimensional hexagonal network in which the carbon atoms form vertices with sp2 hybridisation. Graphene has many exceptional properties, including mechanical strength, optical transparency, and electric and thermal conductivity. Furthermore, the one-atomic layer structure of graphene makes it ultralight and super thin. Graphene has a thickness of about 0.35 nm, which is approximately 1/200,000th of the thickness of human hair. However, the closely arranged carbon atoms and the sp2 orbital hybridisation provide exceptional stability to the graphene structure. Thus, graphene shows extraordinary transparency of 97.7 percent, which means that it only absorbs 2.3 percent of visible light [22].
Due to its high conductivity, graphene can provide a possible alternative to many common substances used as membranes, including indium tin oxide (ITO) and fluorine-doped tin oxide (FTO). The use of graphene for these applications could address the issues of limited indium resources, pollution, and fragility. A membrane with graphene as the primary component could be used as a window barrier in dye-sensitised LEDs and solar cells. It is also possible for graphene to also exist as a nanoribbon in which a lateral charge movement causes an energy barrier to form close to the central point. A reduction in the thickness of the nanoribbon raises this energy barrier. [23]. Hence, by carefully adjusting the width of the graphene nanoribbon, the energy barrier can be accurately regulated, which is a promising advantage for graphene-based electronic devices. Graphene can also be used in the partial detection of external magnetic fields, electric fields, and deformations due to it being a low-noise electrical substance [24].
In terms of its structure, graphene can be considered the basic unit of graphite, fullerene [25], carbon nanotube [26], graphyne [27], and other related materials such as amorphous carbon, carbon fibre, charcoal [28], as well as aromatic molecules such as polycyclic aromatic hydrocarbons. As they all have the same structure, they all have some properties in common, even though their different sizes and shapes make them very different. Thus, the structural study of graphene helps to understand the above-listed materials. It can be observed from Figure 1 that the ( 7 , 7 )   β -Graphene contains seven graphene layers stacked next to each other. In three dimensions, the ( 7 , 7 )   β -Graphene contains seven layers stacked on top of each other. Thus, it can be inferred that the ( m , n )   β -Graphene consists of vertex set V ( Γ ) = 12 m n + 2 m + 10 n and edge set E ( Γ ) = 18 m n + m + 11 n .
Theorem 1.
If Γ is a β-Graphene system, then N M -polynomial of Γ is given as follows
N M ( Γ ; 𝓇 , 𝓉 ) = ( 4 m + 2 n ) 𝓇 5 𝓉 5 + ( 4 m + 8 ) 𝓇 5 𝓉 7 + ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + ( 2 m + 4 ) 𝓇 7 𝓉 9 + ( 2 n 2 ) 𝓇 8 𝓉 8 + ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9
Theorem 2.
If Γ is a β-Graphene system, then N M -polynomial of Γ is given as follows
1.
N M M 1 ( Γ ) = 324 m n 70 m + 154 n ,
2.
N M M 2 ( Γ ) = 1458 m n 599 m + 545 n + 30 ,
3.
N M M 2 m ( Γ ) = ( 2437 9450 ) m + ( 16489 64800 ) n + ( 2 9 ) m n + 1597 90720 ,
4.
N M R α ( Γ ) = 25 φ ( 4 m + 2 n ) + 35 φ ( 4 m + 8 ) + 40 φ ( 4 m + 4 n 8 ) + 63 φ ( 2 m + 4 ) + 64 φ ( 2 n 2 ) + 72 φ ( 8 m + 4 n 12 ) + 81 φ ( 18 m n 21 m n + 10 ) ,
5.
N M R R α ( Γ ) = 1 / 25 φ ( 4 m + 2 n ) + 1 / 35 φ ( 4 m + 8 ) + 1 / 40 φ ( 4 m + 4 n 8 ) + 1 / 63 φ ( 2 m + 4 ) + 1 / 64 φ ( 2 n 2 ) + 1 / 72 φ ( 8 m + 4 n 12 ) + 1 / 81 φ ( 18 m n 21 m n + 10 ) ,
6.
N M S D D ( Γ ) = ( 151 42 ) m + ( 1033 45 ) n + 36 m n 503 630 ,
7.
N M H ( Γ ) = ( 12463 13260 ) m + ( 64637 39780 ) n + 413 7956 + 2 m n ,
8.
N M I ( Γ ) = ( 16685 442 ) n + 81 m n ( 99547 5304 ) m + 1709 2652 ,
9.
N M A ( Γ ) = ( 176959019139103 233744896000 ) n + ( 4782969 2048 ) m n ( 798535392483 681472000 ) m + 11934914516533 116872448000
10.
NMABC ( Γ ) = 34 15 2 m + 4 5 2 n + 4 7 2 7 m + 25 28 2 7 + 1 5 11 2 5 m + 1 5 11 2 5 n 2 5 11 2 5 + 4 3 2 + 1 4 2 7 n + 2 3 5 3 2 m + ( 1 / 3 ) 5 3 2 n 5 3 2 + 8 m n 28 3 m 4 9 n + 40 9
11.
N M G A ( Γ ) = 3 n + 2 3 7 5 m + 4 3 7 5 + 16 13 2 5 m + 16 13 2 5 n 32 13 2 5 + 3 4 7 m + 3 2 7 + 8 + 96 17 2 m + 48 17 2 n 144 17 2 + 18 m n 17 m ,
12.
N M B 1 ( Γ ) = 900 m n 214 m + 418 n
13.
N M B 2 ( Γ ) = 5184 m n 2188 m + 1912 n + 84 ,
14.
N M H B 1 ( Γ ) = 746496 m n 473476 m + 188680 n + 4801
Proof of Theorem 2.
Let f ( 𝓇 , 𝓉 ) = N M ( Γ ; 𝓇 , 𝓉 ) = ( 4 m + 2 n ) 𝓇 5 𝓉 5 + ( 4 m + 8 ) 𝓇 5 𝓉 7 + ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + ( 2 m + 4 ) 𝓇 7 𝓉 9 + ( 2 n 2 ) 𝓇 8 𝓉 8 + ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9
D 𝓇 (   f ( 𝓇 , 𝓉 ) ) = 5 ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 5 ( 4 m + 8 ) 𝓇 5 𝓉 7 + 5 ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 7 ( 2 m + 4 ) 𝓇 7 𝓉 9 + 8 ( 2 n 2 ) 𝓇 8 𝓉 8 + 8 ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 9 ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
D 𝓉 (   f ( 𝓇 , 𝓉 ) ) = 5 ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 7 ( 4 m + 8 ) 𝓇 5 𝓉 7 + 8 ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 9 ( 2 m + 4 ) 𝓇 7 𝓉 9 + 8 ( 2 n 2 ) 𝓇 8 𝓉 8 + 8 ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 9 ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
D 𝓇 + D 𝓉 (   f ( 𝓇 , 𝓉 ) ) = 10 ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 12 ( 4 m + 8 ) 𝓇 5 𝓉 7 + 13 ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 16 ( 2 m + 4 ) 𝓇 7 𝓉 9 + 16 ( 2 n 2 ) 𝓇 8 𝓉 8 + 17 ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 18 ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
D 𝓇 D 𝓉 (   f ( 𝓇 , 𝓉 ) ) = 25 ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 35 ( 4 m + 8 ) 𝓇 5 𝓉 7 + 40 ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 63 ( 2 m + 4 ) 𝓇 7 𝓉 9 + 64 ( 2 n 2 ) 𝓇 8 𝓉 8 + 72 ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 81 ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
S 𝓇 (   f ( 𝓇 , 𝓉 ) ) = 1 5 ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 1 5 ( 4 m + 8 ) 𝓇 5 𝓉 7 + 1 5 ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 1 7 ( 2 m + 4 ) 𝓇 7 𝓉 9 + 1 8 ( 2 n 2 ) 𝓇 8 𝓉 8 + 1 8 ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 1 9 ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
S 𝓉 (   f ( 𝓇 , 𝓉 ) ) = 1 5 ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 1 7 ( 4 m + 8 ) 𝓇 5 𝓉 7 + 1 8 ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 1 9 ( 2 m + 4 ) 𝓇 7 𝓉 9 + 1 8 ( 2 n 2 ) 𝓇 8 𝓉 8 + 1 9 ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 1 9 ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
S 𝓉 S 𝓇 ( f ( 𝓇 , 𝓉 ) ) = 1 25 ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 1 35 ( 4 m + 8 ) 𝓇 5 𝓉 7 + 1 40 ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 1 63 ( 2 m + 4 ) 𝓇 7 𝓉 9 + 1 64 ( 2 n 2 ) 𝓇 8 𝓉 8 + 1 72 ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 1 81 ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
D 𝓇 φ D 𝓉 φ (   f ( 𝓇 , 𝓉 ) ) = 25 φ ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 35 φ ( 4 m + 8 ) 𝓇 5 𝓉 7 + 40 φ ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 63 φ ( 2 m + 4 ) 𝓇 7 𝓉 9 + 64 φ ( 2 n 2 ) 𝓇 8 𝓉 8 + 72 φ ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 81 φ ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
S 𝓇 φ S 𝓉 φ ( f ( 𝓇 , 𝓉 ) ) = 1 25 φ ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 1 35 φ ( 4 m + 8 ) 𝓇 5 𝓉 7 + 1 40 φ ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 1 63 φ ( 2 m + 4 ) 𝓇 7 𝓉 9 + 1 64 φ ( 2 n 2 ) 𝓇 8 𝓉 8 + 1 72 φ ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 1 81 φ ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
S 𝓉 D 𝓇 (   f ( 𝓇 , 𝓉 ) ) = 1 ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 5 7 ( 4 m + 8 ) 𝓇 5 𝓉 7 + 5 8 ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 7 9 ( 2 m + 4 ) 𝓇 7 𝓉 9 + 1 ( 2 n 2 ) 𝓇 8 𝓉 8 + 8 9 ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 1 ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
S 𝓇 D 𝓉 (   f ( 𝓇 , 𝓉 ) ) = 1 ( 4 m + 2 n ) 𝓇 5 𝓉 5 + 7 5 ( 4 m + 8 ) 𝓇 5 𝓉 7 + 8 5 ( 4 m + 4 n 8 ) 𝓇 5 𝓉 8 + 9 7 ( 2 m + 4 ) 𝓇 7 𝓉 9 + 1 ( 2 n 2 ) 𝓇 8 𝓉 8 + 9 8 ( 8 m + 4 n 12 ) 𝓇 8 𝓉 9 + 1 ( 18 m n 21 m n + 10 ) 𝓇 9 𝓉 9 ,
J (   f ( 𝓇 , 𝓉 ) ) = f ( 𝓇 , 𝓇 ) = ( 4 m + 2 n ) 𝓇 10 + ( 4 m + 8 ) 𝓇 12 + ( 4 m + 4 n 8 ) 𝓇 13 + ( 2 m + 4 ) 𝓇 16 + ( 2 n 2 ) 𝓇 16 + ( 8 m + 4 n 12 ) 𝓇 17 + 9 ( 18 m n 21 m n + 10 ) 𝓇 18 ,
S 𝓇 J   f ( 𝓇 , 𝓉 ) = 1 10 ( 4 m + 2 n ) 𝓇 10 + 1 12 ( 4 m + 8 ) 𝓇 12 + 1 13 ( 4 m + 4 n 8 ) 𝓇 13 + 1 16 ( 2 m + 4 ) 𝓇 16 + 1 16 ( 2 n 2 ) 𝓇 16 + 1 17 ( 8 m + 4 n 12 ) 𝓇 17 + 1 18 ( 18 m n 21 m n + 10 ) 𝓇 18 ,
S 𝓇 J   D 𝓇 D 𝓉 f ( 𝓇 , 𝓉 ) = 25 10 ( 4 m + 2 n ) 𝓇 10 + 35 12 ( 4 m + 8 ) 𝓇 12 + 40 13 ( 4 m + 4 n 8 ) 𝓇 13 + 63 16 ( 2 m + 4 ) 𝓇 16 + 64 16 ( 2 n 2 ) 𝓇 16 + 72 17 ( 8 m + 4 n 12 ) 𝓇 17 + 81 18 ( 18 m n 21 m n + 10 ) 𝓇 18 ,
S 𝓇 3 Q 2 J   D 𝓇 3 D 𝓉 3 f ( 𝓇 , 𝓉 ) = ( 25 8 ) 3 ( 4 m + 2 n ) 𝓇 8 + ( 7 2 ) 3 ( 4 m + 8 ) 𝓇 10 + ( 40 11 ) 3 ( 4 m + 4 n 8 ) 𝓇 11 + ( 9 2 ) 3 ( 2 m + 4 ) 𝓇 14 + ( 32 7 ) 3 ( 2 n 2 ) 𝓇 14 + ( 24 5 ) 3 ( 8 m + 4 n 12 ) 𝓇 15 + ( 81 16 ) 3 ( 18 m n 21 m n + 10 ) 𝓇 16 ,
D 𝓇 1 2 Q 2 J   S 𝓇 1 2 S 𝓉 1 2 f ( 𝓇 , 𝓉 ) = 8 5 ( 4 m + 2 n ) 𝓇 8 + 10 35 ( 4 m + 8 ) 𝓇 10 + 11 40 ( 4 m + 4 n 8 ) 𝓇 11 + 14 3 7 ( 2 m + 4 ) 𝓇 14 + 14 8 ( 2 n 2 ) 𝓇 14 + 15 3 8 ( 8 m + 4 n 12 ) 𝓇 15 + 16 9 ( 18 m n 21 m n + 10 ) 𝓇 16 ,
2 S 𝓇 J D 𝓇 1 2 D t 1 2 f ( 𝓇 , 𝓉 ) = 10 10 ( 4 m + 2 n ) 𝓇 10 + 2 35 12 ( 4 m + 8 ) 𝓇 12 + 2 40 13 ( 4 m + 4 n 8 ) 𝓇 13 + 2 63 16 ( 2 m + 4 ) 𝓇 16 + 16 16 ( 2 n 2 ) 𝓇 16 + 2 72 17 ( 8 m + 4 n 12 ) 𝓇 17 + 18 18 ( 18 m n 21 m n + 10 ) 𝓇 18 ,
2 D 𝓇 Q 2 J f ( 𝓇 , 𝓉 ) = 2 · 8 ( 4 m + 2 n ) 𝓇 8 + 2 · 10 ( 4 m + 8 ) 𝓇 10 + 2 · 11 ( 4 m + 4 n 8 ) 𝓇 11 + 2 · 14 ( 2 m + 4 ) 𝓇 14 + 2 · 14 ( 2 n 2 ) 𝓇 14 + 2 · 15 ( 8 m + 4 n 12 ) 𝓇 15 + 2 · 16 ( 18 m n 21 m n + 10 ) 𝓇 16 ,
D 𝓇 Q 2 J ( D 𝓇 + D 𝓉 ) f ( 𝓇 , 𝓉 ) = 8 · 10 ( 4 m + 2 n ) 𝓇 8 + 10 · 12 ( 4 m + 8 ) 𝓇 10 + 11 · 13 ( 4 m + 4 n 8 ) 𝓇 11 + 14 · 16 ( 2 m + 4 ) 𝓇 14 + 14 · 16 ( 2 n 2 ) 𝓇 14 + 15 · 17 ( 8 m + 4 n 12 ) 𝓇 15 + 16 · 18 ( 18 m n 21 m n + 10 ) 𝓇 16 ,
D 𝓇 2 Q 2 J ( D 𝓇 2 + D 𝓉 2 ) f ( 𝓇 , 𝓉 ) = 64 · 50 ( 4 m + 2 n ) 𝓇 8 + 100 · 74 ( 4 m + 8 ) 𝓇 10 + 121 · 89 ( 4 m + 4 n 8 ) 𝓇 11 + 196 · 130 ( 2 m + 4 ) 𝓇 14 + 196 · 128 ( 2 n 2 ) 𝓇 14 + 225 · 145 ( 8 m + 4 n 12 ) 𝓇 15 + 256 · 162 ( 18 m n 21 m n + 10 ) 𝓇 16 .
The above results are obtained by using the conditions of M-Polynomial with its derivatives, and the partition Table 2. □

5. Neighbourhood Degree Sum-Based Entropy Measures for β-Graphene

Theorem 3.
If Γ is a β-Graphene system, then N M -entropy measures of Γ are given as follows
1.
N E N T N M 1 ( Γ ) = log ( 324 m n 70 m + 154 n ) 1 324 m n 70 m + 154 n [ ( 4 m + 2 n ) ( 10 ) log 10 + ( 4 m + 8 ) ( 12 ) log 12 + ( 4 m + 4 n 8 ) ( 13 ) log 13 + ( 2 m + 4 ) ( 16 ) log 16 + ( 2 n 2 ) ( 16 ) log 16 + ( 8 m + 4 n 12 ) ( 17 ) log 17 + ( 18 m n 21 m n + 10 ) ( 18 ) log 18 ]
= l o g ( 324 m n 70 m + 154 n ) ( 273.7788 m + 408.7804 n + 2.8232 + 936.4896 m n 324 m n 70 m + 154 n )
2.
N E N T N M 2 ( Γ ) = log ( 1458 m n 599 m + 545 n + 30 ) 1 1458 m n 599 m + 545 n + 30 [ ( 4 m + 2 n ) ( 25 ) log 25 + ( 4 m + 8 ) ( 35 ) log 35 + ( 4 m + 4 n 8 ) ( 40 ) log 40 + ( 2 m + 4 ) ( 63 ) log 63 + ( 2 n 2 ) ( 64 ) log 64 + ( 8 m + 4 n 12 ) ( 72 ) log 72 + ( 18 m n 21 m n + 10 ) ( 81 ) log 81 ]
= l o g ( 1458 m n 599 m + 545 n + 30 ) ( 3079.6186 m + 2159.2464 n + 191.1532 + 6407.0352 m n 1458 m n 599 m + 545 n + 30 )
3.
NNET N M 2 m ( Γ ) = log 2437 9450 m + 16489 64800 n + 2 9 m n + 1597 90720 1 2437 9450 m + 16489 64800 n + 2 9 m n + 1597 90720 ( 4 m + 2 n ) 1 25 log 1 25 + ( 4 m + 8 ) 1 35 log 1 35 + ( 4 m + 4 n 8 ) 1 40 log 1 40 + ( 2 m + 4 ) 1 63 log 1 63 + ( 2 n 2 ) 1 64 log 1 64 + ( 8 m + 4 n 12 ) 1 72 log 1 72 + ( 18 m n 21 m n + 10 ) 1 81 log 1 81
= l o g ( ( 2437 9450 ) m + ( 16489 64800 ) n + ( 2 9 ) m n + 1597 90720 ) ( .7576449840 m .9397022176 n 0.376835284 e 1 .9765333333 m n ( 2437 9450 ) m + ( 16489 64800 ) n + ( 2 9 ) m n + 1597 90720 )
4.
N E N T N H M ( Γ ) = log ( 5832 m n 2328 m + 2220 n + 84 ) 1 5832 m n 2328 m + 2220 n + 84 [ ( 4 m + 2 n ) ( 100 ) log 100 + ( 4 m + 8 ) ( 144 ) log 144 + ( 4 m + 4 n 8 ) ( 169 ) log 169 + ( 2 m + 4 ) ( 256 ) log 256 + ( 2 n 2 ) ( 256 ) log 256 + ( 8 m + 4 n 12 ) ( 289 ) log 289 + ( 18 m n 21 m n + 10 ) ( 324 ) log 324 ]
= l o g ( 5832 m n 2328 m + 2220 n + 84 ) ( 15220.2744 m + 11905.3064 n + 707.5792 + 33713.6256 m n 5832 m n 2328 m + 2220 n + 84 )
5.
N E N T N A ( Γ ) = log 4782969 2048 m n 798535392483 681472000 m + 176959019139103 233744896000 n + 11934914516533 116872448000 1 4782969 2048 m n 798535392483 681472000 m + 176959019139103 233744896000 n + 11934914516533 116872448000 [ ( 4 m + 2 n ) 25 8 3 log 25 8 3 + ( 4 m + 8 ) 35 10 3 log 35 3 10 + ( 4 m + 4 n 8 ) 40 11 3 log 40 3 11 + ( 2 m + 4 ) 63 14 3 log 63 3 14 + ( 2 n 2 ) 64 14 3 log 64 3 14 + ( 8 m + 4 n 12 ) 72 15 3 log 72 3 15 + ( 18 m n 21 m n + 10 ) 81 16 3 log 81 3 16
= l o g ( ( 4782969 2048 ) m n ( 798535392483 681472000 ) m + ( 176959019139103 233744896000 ) n + 11934914516533 116872448000 ) 6658.292895 m + 2328.656416 n + 829.875339 + 9964.129511 m n ( 4782969 2048 ) m n ( 798535392483 681472000 ) m + ( 176959019139103 233744896000 ) n + 11934914516533 116872448000
6.
N E N T N A B C ( Γ ) = log ( N A B C ( Γ ) ) 1 N A B C ( Γ ) ( 4 m + 2 n ) 8 25 log 8 25 + ( 4 m + 8 ) 10 35 log 10 35 + ( 4 m + 4 n 8 ) 11 40 log 11 40 + ( 2 m + 4 ) 14 63 log 14 63 + ( 2 n 2 ) 14 64 log 14 64 + ( 8 m + 4 n 12 ) 15 72 log 15 72 + ( 18 m n 21 m n + 10 ) 16 81 log 16 81
7.
N E N T N G A ( Γ ) = log ( N G A ( Γ ) ) 1 N G A ( Γ ) ( 4 m + 2 n ) ( 1 ) log 1 + ( 4 m + 8 ) 2 35 12 log 2 35 12 + ( 4 m + 4 n 8 ) 2 40 13 log 2 40 13 + ( 2 m + 4 ) 2 63 16 log 2 63 16 + ( 2 n 2 ) ( 1 ) log 1 + ( 8 m + 4 n 12 ) 2 72 17 log 2 72 17 + ( 18 m n 21 m n + 10 ) ( 1 ) log 1
8.
NENT N A G 1 ( Γ ) = log A G 1 ( Γ ) 1 A G 1 ( Γ ) ( 4 m + 2 n ) ( 1 ) log 1 + ( 4 m + 8 ) 12 2 35 log 12 2 35 + ( 4 m + 4 n 8 ) 13 2 40 log 13 2 40 + ( 2 m + 4 ) 16 2 63 log 16 2 63 + ( 2 n 2 ) ( 1 ) log 1 + ( 8 m + 4 n 12 ) 17 2 72 log 17 2 72 + ( 18 m n 21 m n + 10 ) ( 1 ) log 1
9.
N E N T N F 1 ( Γ ) = log ( 2916 m n 1130 m + 1130 n + 24 ) 1 2916 m n 1130 m + 1130 n + 24 [ ( 4 m + 2 n ) ( 50 ) log 50 + ( 4 m + 8 ) ( 74 ) log 74 + ( 4 m + 4 n 8 ) ( 89 ) log 89 + ( 2 m + 4 ) ( 130 ) log 130 + ( 2 n 2 ) ( 128 ) log 128 + ( 8 m + 4 n 12 ) ( 145 ) log 145 + ( 18 m n 21 m n + 10 ) ( 162 ) log 162 ]
= log ( 2916 m n 1130 m + 1130 n + 24 ) 6615.1380 m + 5293.5484 n + 223.5860 + 14835.4416 m n 2916 m n 1130 m + 1130 n + 24
10.
N E N T N F 2 ( Γ ) = log ( 118098 m n 74571 m + 30017 n + 8086 ) 1 118098 m n 74571 m + 30017 n + 8086 [ ( 4 m + 2 n ) ( 625 ) log 625 + ( 4 m + 8 ) ( 1225 ) log 1225 + ( 4 m + 4 n 8 ) ( 1600 ) log 1600 + ( 2 m + 4 ) ( 3969 ) log 3969 + ( 2 n 2 ) ( 4096 ) log 4096 + ( 8 m + 4 n 12 ) ( 5184 ) log 5184 + ( 18 m n 21 m n + 10 ) ( 6561 ) log 6561 ]
= l o g ( 118098 m n 74571 m + 30017 n + 8086 ) ( 6.922733326 × 10 5 m + 2.431043232 × 10 5 n + 83203.5944 + 1.037939702 × 10 6 m n 118098 m n 74571 m + 30017 n + 8086 )
11.
N E N T N χ ( Γ ) = log ( N χ ( Γ ) ) 1 N χ ( Γ ) ( 4 m + 2 n ) 1 10 log 1 10 + ( 4 m + 8 ) 1 12 log 1 12 + ( 4 m + 4 n 8 ) 1 13 log 1 13 + ( 2 m + 4 ) 1 16 log 1 16 + ( 2 n 2 ) 1 16 log 1 16 + ( 8 m + 4 n 12 ) 1 17 log 1 17 + ( 18 m n 21 m n + 10 ) 1 18 log 1 18
12.
N E N T N ReZ Γ 1 ( Γ ) = log 146 35 m + 274 35 n + 516 35 m n 1 146 35 m + 274 35 n + 516 35 m n [ ( 4 m + 2 n ) 10 25 log 10 25 + ( 4 m + 8 ) 12 35 log 12 35 + ( 4 m + 4 n 8 ) 13 40 log 13 40 + ( 2 m + 4 ) 16 63 log 16 63 + ( 2 n 2 ) 16 64 log 16 64 + ( 8 m + 4 n 12 ) 17 72 log 17 72 + ( 18 m n 21 m n + 10 ) 18 81 log 18 81
= l o g ( ( 146 35 ) m + ( 274 35 ) n + ( 516 35 ) m n ) ( .798715906 m 3.916313112 n + 0.34553969 e 1 6.016399999 m n ( 146 35 ) m + ( 274 35 ) n + ( 516 35 ) m n )
13.
N E N T N R e Z G 2 ( Γ ) = log 1309 18 n + 121 m n 451 18 m 1 1309 18 n + 121 m n 451 18 m ( 4 m + 2 n ) 25 10 log 25 10 + ( 4 m + 8 ) 35 12 log 35 12 + ( 4 m + 4 n 8 ) 40 13 log 40 13 + ( 2 m + 4 ) 63 16 log 63 16 + ( 2 n 2 ) 64 16 log 64 16 + ( 8 m + 4 n 12 ) 72 17 log 72 17 + ( 18 m n 21 m n + 10 ) 81 18 log 81 18
= l n ( ( 1309 18 ) n + 121 m n ( 451 18 ) m ) ( 46.95207065 m + 47.19060362 n + 2.12647952 + 121.8321000 m n ( 1309 18 ) n + 121 m n ( 451 18 ) m )
14.
N E N T N R e Z G 3 ( Γ ) = log ( 26244 m n 14050 m + 8066 n + 1076 ) 1 26244 m n 14050 m + 8066 n + 1076 [ ( 4 m + 2 n ) ( 250 ) log 250 + ( 4 m + 8 ) ( 420 ) log 420 + ( 4 m + 4 n 8 ) ( 520 ) log 520 + ( 2 m + 4 ) ( 1008 ) log 1008 + ( 2 n 2 ) ( 1024 ) log 1024 + ( 8 m + 4 n 12 ) ( 1224 ) log 1224 + ( 18 m n 21 m n + 10 ) ( 1458 ) log 1458 ]
= l o g ( 26244 m n 14050 m + 8066 n + 1076 ) ( 1.108067064 × 10 5 m + 54153.1980 n + 9750.1632 + 1.911822912 × 10 5 m n 26244 m n 14050 m + 8066 n + 1076 )
where N A B C ( Γ ) , N M G A ( Γ ) , N A G 1 ( Γ ) and N χ ( Γ ) can be represented as N A B C ( Γ ) = ( 34 15 ) 2 m + ( 4 5 ) 2 n + ( 4 7 ) 2 7 m + ( 25 28 ) 2   7 + ( 1 5 ) 2   5   11 m + ( 1 5 ) 2 5   11 n ( 2 5 ) 2 5   11 + ( 4 3 ) 2 + ( 1 4 ) 2 7 n + ( 2 3 ) 2 5 3 m + ( 1 3 ) 2 5   3 n s 2 5 3 + 8 m n ( 28 3 ) m ( 4 9 ) n + 40 9 , N M G A ( Γ ) = 18 m n 17 m + 3 n + ( 2 3 ) 7 5 m + ( 4 3 ) 7 5 + ( 16 13 ) 2 5 m + ( 16 13 ) 2 5 n ( 32 13 ) 2 5 + ( 3 4 ) 7 m + ( 3 2 ) 7 + 8 + ( 96 17 ) 2 m + ( 48 17 ) 2   n ( 144 17 ) 2 , N M A G 1 = 18 m n 17 m + 3 n + ( 24 35 ) 7 5 m + ( 48 35 ) 7 5 + ( 13 10 ) 2 5 m + ( 13 10 ) 2 5 n ( 13 5 ) 2 5 + ( 16 21 ) 7 m + ( 32 21 ) 7 + 8 + ( 17 3 ) 2 m + ( 17 6 ) 2 n ( 17 2 ) 2 and N M χ ( Γ ) = ( 2 5 ) 2 5 m + ( 1 5 ) 2 5 n + ( 2 3 ) 3 m + ( 4 3 ) 3 + ( 4 13 ) 13 m + ( 4 13 ) 13 n ( 8 13 ) 13 + ( 1 2 ) m + 1 2 + ( 1 2 ) n + ( 8 17 ) 17 m + ( 4 17 ) 17 n ( 12 17 ) 17 + 3 2 m n ( 7 2 ) 2 m ( 1 6 ) 2 n + ( 5 3 ) 2 .

6. Comparative Analysis for β-Graphene

In this section, the analytical expressions of the neighbourhood degree sum-based indices derived from Theorem 2 and Theorem 3 are represented as 3D plots. These plots help the reader visually interpret and understand the behaviour of the indices with respect to the variables that define the molecular structure. In addition, the results of Theorem 2 have also been represented as comparison plots where all the neighbourhood degree sum-based indices are plotted in the same graph against the same structural variables. These comparison plots provide a graphical representation of how the indices vary with respect to each other and the molecular structure. These plots are presented in Figure 2 and Figure 3. The numerical values of the indices are computed and listed in Table 3, Table 4, Table 5 and Table 6. The numerical values are represented as 3D plots in Figure 3, Figure 4, Figure 5, and Figure 6 respectively to provide a visual understanding to the readers.

7. Neighbourhood Degree Sum-Based M-Polynomial for β-Graphyne

Graphyne is a type of graphene in which the hexagons are connected by acetylenic bonds, as shown in Figure 1. Graphyne is especially intriguing because its distinct electronic structure sets it apart from other commonly available carbon allotropes, such as diamond and graphite. Furthermore, adding transition metals to graphyne sheets significantly affects the material’s general behaviour [29]. For example, the adsorption of chromium and iron transforms graphyne from a semiconductor into a spin-polarised metal. Other transition metal adsorption can produce a spin-polarised half semiconductor or a narrow gap semiconductor.
Computational studies demonstrate that the band gap of graphyne is mechanically adjustable, which facilitates the fabrication of transistors with various characteristics that are directly linked to the band gap. It also allows for the determination of these properties much later in the manufacturing process when compared to conventional semiconductors, where the properties are controlled based on doping [30,31,32].
Moreover, research has revealed that graphyne possesses peculiar optical properties. Although graphyne has not yet been successfully synthesised, its properties can be used in a variety of applications, including anisotropic conductors, nanofillers, desalinators, semiconductor metal hybrids, transistors, and sensors. The mechanical properties of graphyne enable its use as a nanofiller in composite materials. A sizeable elastic strain region of graphyne also makes it possible to drastically alter the material and revert it to its original shape without lasting damage [32]. This property allows for resilient electromechanical coupling, which is advantageous for many applications, including temperature sensing. The cardinality of β -Graphynes is V ( Γ ) = 36 m n + 2 m + 24 n and E ( Γ ) = 42 m n + m + 23 n . It is represented in Figure 7. The edge partitions for the β -Graphynes are presented in Table 7.
Theorem 4.
If Γ is a β- graphyne system, then N M -polynomial of Γ is given as follows
N M ( Γ ; 𝓇 , 𝓉 ) = ( 12 m + 6 n ) 𝓇 4 𝓉 4 + ( 8 m + 4 n ) 𝓇 4 𝓉 5                                                           + ( 12 m n 2 m 4 n ) 𝓇 5 𝓉 5 + ( 24 m n + 4 m 4 n ) 𝓇 5 𝓉 7                                                           + ( 6 m n 21 m + 21 n ) 𝓇 7 𝓉 7
Proof. 
Let f ( 𝓇 , 𝓉 ) = N M ( Γ ; 𝓇 , 𝓉 ) = ( 12 m + 6 n ) 𝓇 4 𝓉 4 + ( 8 m + 4 n ) 𝓇 4 𝓉 5 + ( 12 m n 2 m 4 n ) 𝓇 5 𝓉 5 + ( 24 m n + 4 m 4 n ) 𝓇 5 𝓉 7 + ( 6 m n 21 m + 21 n ) 𝓇 7 𝓉 7
Theorem 5.
If Γ is a β- graphyne system, then N M -polynomial of Γ is given as follows
1.
NM 1 ( Γ ) = 492 mn 98 m + 290 n
2.
NM 2 ( Γ ) = 1434 mn 587 m + 965 n ,
3.
NM 2 m ( Γ ) = ( 529 700 ) m + ( 1021 1400 ) n + ( 1578 1225 ) mn ,
4.
NR α ( Γ ) = 16 φ ( 12 m + 6 n ) + 20 φ ( 8 m + 4 n ) + 25 φ ( 12 mn 2 m 4 n ) + 35 φ ( 24 mn + 4 m 4 n ) + 49 φ ( 6 mn 21 m + 21 n ) ,
5.
NRR α ( Γ ) = 1 16 φ ( 12 m + 6 n ) + 1 20 φ ( 8 m + 4 n ) + 1 25 φ ( 12 mn 2 m 4 n ) + 1 35 φ ( 24 mn + 4 m 4 n ) + 1 49 φ ( 6 mn 21 m + 21 n ) ,
6.
NSDD ( Γ ) = ( 20 7 ) m + ( 1601 35 ) n + ( 3036 35 ) mn ,
7.
NH ( Γ ) = ( 92 45 ) m + ( 353 90 ) n + ( 254 35 ) mn ,
8.
NI ( Γ ) = 121 mn ( 451 18 ) m + ( 1309 18 ) n ,
9.
NA ( Γ ) = ( 2077879 1152 ) mn ( 715316939 790272 ) m + ( 530771995 395136 ) n ,
10.
NABC ( Γ ) = 3 2 3 m + 3 2 2 3 n + 4 5 7 5 m + 2 5 7 5 n + 24 5 2 mn 4 5 2 m 8 5 2 n + 24 7 2 7 m n + 4 7 2 7 m 4 7 2 7 n + 12 7 3 mn 6 3 m + 6 3 n ,
11.
NGA ( Γ ) = 23 n + ( 32 9 ) 5 m + 11 m + ( 16 9 ) 5 n + 18 mn + 4 7   5 mn + ( 2 3 ) 7   5 m ( 2 3 ) 7   5 n ,
12.
NB 1 ( Γ ) = 1308 mn 298 m + 778 n ,
13.
NB 2 ( Γ ) = 4848 mn 2128 m + 3268 n ,
14.
NHB 1 ( Γ ) = 300672 mn 243256 m + 268900 n .

8. Computing Neighbourhood Degree Sum-Based Entropy Measures for β-graphyne

Theorem 6.
If Γ is a β-Graphyne system, then N M -entropy measures of Γ are given as follows
1.
N E N T N M 1 ( Γ ) = log ( 492 m n 98 m + 290 n ) 1 492 m n 98 m + 290 n [ ( 12 m + 6 n ) ( 8 ) log ( 8 ) + ( 8 m + 4 n ) ( 9 ) log 9 + ( 12 m n 2 m 4 n ) ( 10 ) log 10 + ( 24 m n + 4 m 4 n ) ( 12 ) log 12 + ( 6 m n 21 m + 21 n ) ( 14 ) log 14 ]
= l o g ( 492 m n 98 m + 290 n ) ( 344.8514 m + 743.4266 n + 1213.6476 m n 492 m n 98 m + 290 n )
2.
N E N T N M 2 m ( Γ ) = log ( 1434 m n 587 m + 965 n ) 1 1434 m n 587 m + 965 n [ ( 12 m + 6 n ) ( 16 ) log 16 + ( 8 m + 4 n ) ( 20 ) log 20 + ( 12 m n 2 m 4 n ) ( 25 ) log 25 + ( 24 m n + 4 m 4 n ) ( 35 ) log 35 + ( 6 m n 21 m + 21 n ) ( 49 ) log 49 ]
= l o g ( 1434 m n 587 m + 965 n ) ( 2656.2090 m + 3690.8658 n + 5096.2812 m n 1434 m n 587 m + 965 n )
3.
N E N T N M 2 m ( Γ ) = log 529 700 m + 1021 1400 n + 1578 1225 m n 1 529 700 m + 1021 1400 n + 1578 1225 m n ( 12 m + 6 n ) 1 16 log 1 16 + ( 8 m + 4 n ) 1 20 log 1 20 + ( 12 m n 2 m 4 n ) 1 25 log 1 25 + ( 24 m n + 4 m 4 n ) 1 35 log 1 35 + ( 6 m n 21 m + 21 n ) 1 49 log 1 49
= l o g ( ( 529 700 ) m + ( 1021 1400 ) n + ( 1578 1225 ) m n ) ( 1.758631714 m 2.385451286 n 4.459490939 m n ( 529 700 ) m + ( 1021 1400 ) n + ( 1578 1225 ) m n )
4.
N E N T N H M ( Γ ) = log ( 5832 m n 2328 m + 2220 n + 84 ) 1 5832 m n 2328 m + 2220 n + 84 [ ( 12 m + 6 n ) ( 64 ) log 64 + ( 8 m + 4 n ) ( 81 ) log 81 + ( 12 m n 2 m 4 n ) ( 100 ) log 100 + ( 24 m n + 4 m 4 n ) ( 144 ) log 144 + ( 6 m n 21 m + 21 n ) ( 196 ) log 196 ]
= l o g ( 5832 m n 2328 m + 2220 n + 84 ) ( 13741.9000 m + 20041.1896 n + 28909.0320 m n 5832 m n 2328 m + 2220 n + 84 )
5.
N E N T N A ( Γ ) = log 2077879 1152 m n + 530771995 395136 n 715316939 790272 m 1 2077879 1152 m n + 530771995 395136 n 715316939 790272 m ( 12 m + 6 n ) 16 6 3 log 16 3 6 + ( 8 m + 4 n ) 20 7 3 log 20 7 3 + ( 12 m n 2 m 4 n ) 25 8 3 log 25 3 8 + ( 24 m n + 4 m 4 n ) 35 10 3 log 35 3 10 + ( 6 m n 21 m + 21 n ) 49 12 3 log 49 3 12
= l o g ( ( 2077879 1152 ) m n + ( 530771995 395136 ) n ( 715316939 790272 ) m ) ( 3304.057315 m + 3679.118326 n + 3702.622388 m n ( 2077879 1152 ) m n + ( 530771995 395136 ) n ( 715316939 790272 ) m )
6.
N E N T N A B C ( Γ ) = log ( 5832 m n 2328 m + 2220 n + 84 ) 1 5832 m n 2328 m + 2220 n + 84 ( 12 m + 6 n ) 6 16 log 6 16 + ( 8 m + 4 n ) 7 20 log 7 20 + ( 12 m n 2 m 4 n ) 8 25 log 8 25 + ( 24 m n + 4 m 4 n ) 10 35 log 10 35 + ( 6 m n 21 m + 21 n ) 12 49 log 12 49
7.
N E N T N G A ( Γ ) = log 23 n + 32 9 5 m + 16 9 5 n + 18 m n + 4 5 7 m n + 2 3 7 5 m 2 3 7 5 n 11 m 1 23 n + 32 9 5 m + 16 9 5 n + 18 m n + 4 5 7 m n + 2 3 7 5 m 2 3 7 5 n 11 m [ ( 12 m + 6 n ) ( 1 ) log 1 + ( 8 m + 4 n ) 2 20 9 log 2 20 9 + ( 12 m n 2 m 4 n ) ( 1 ) log 1 + ( 24 m n + 4 m 4 n ) 2 35 12 log 2 35 12 + ( 6 m n 21 m + 21 n ) ( 1 ) log 1
8.
N E N T N A G 1 ( Γ ) = log 23 n + 18 5 5 m + 9 5 5 n + 18 m n + 144 35 5 7 m n + 24 35 5 7 m 24 35 7 5 n 11 m 1 23 n + 18 5 5 m + 9 5 5 n + 18 m n + 144 35 5 7 m n + 24 35 5 7 m 24 35 7 5 n 11 m [ ( 12 m + 6 n ) ( 1 ) log 1 + ( 8 m + 4 n ) 9 2 20 log 9 2 20 + ( 12 m n 2 m 4 n ) ( 1 ) log 1 + ( 24 m n + 4 m 4 n ) 12 2 35 log 12 2 35 + ( 6 m n 21 m + 21 n ) ( 1 ) log 1
9.
N E N T N F 1 ( Γ ) = log ( 2964 m n 1150 m + 1918 n ) 1 2964 m n 1150 m + 1918 n [ ( 12 m + 6 n ) ( 32 ) log ( 32 ) + ( 8 m + 4 n ) ( 41 ) log 41 + ( 12 m n 2 m 4 n ) ( 50 ) log 50 + ( 24 m n + 4 m 4 n ) ( 74 ) log 74 + ( 6 m n 21 m + 21 n ) ( 98 ) log 98 ]
= l o g ( 2964 m n 1150 m + 1918 n ) ( 3256.8298 m + 5906.6218 n + 11902.3020 m n 2964 m n 1150 m + 1918 n )
10.
N E N T N F 2 ( Γ ) = log ( 49146 m n 32939 m + 38597 n ) 1 49146 m n 32939 m + 38597 n [ ( 12 m + 6 n ) ( 256 ) log ( 256 ) + ( 8 m + 4 n ) ( 400 ) log 400 + ( 12 m n 2 m 4 n ) ( 625 ) log 625 + ( 24 m n + 4 m 4 n ) ( 1225 ) log 1225 + ( 6 m n 21 m + 21 n ) ( 2401 ) log 2401 ]
= log ( 49146 m n 32939 m + 38597 n ) ( 3.294546212 × m + 3.596246228 × 10 5 n + 3.694641816 × 10 5 m n 49146 m n 32939 m + 38597 n )
11.
N E N T N χ ( Γ ) = log ( χ ( Γ ) ) 1 χ ( Γ ) ( 12 m + 6 n ) 1 8 log 1 8 + ( 8 m + 4 n ) 1 9 log 1 9 + ( 12 m n 2 m 4 n ) 1 10 log 1 10 + ( 24 m n + 4 m 4 n ) 1 12 log 1 12 + ( 6 m n 21 m + 21 n ) 1 14 log 1 14
12.
N E N T N R e Z Γ 1 ( Γ ) = log 146 35 m + 274 35 n + 516 35 m n 1 146 35 m + 274 35 n + 516 35 m n ( 12 m + 6 n ) 8 16 log 8 16 + ( 8 m + 4 n ) 9 20 log 9 20 + ( 12 m n 2 m 4 n ) 10 25 log 10 25 + ( 24 m n + 4 m 4 n ) 12 35 log 12 35 + ( 6 m n 21 m + 21 n ) 14 49 log 14 49
= l o g ( ( 146 35 ) m + ( 274 35 ) n + ( 516 35 ) m n ) ( .251681144 m 8.099526856 n 15.35371200 m n ( 146 35 ) m + ( 274 35 ) n + ( 516 35 ) m n )
13.
NENT N NReZG 2 ( Γ ) = log 1309 18 n + 121 m n 451 18 m 1 1309 18 n + 121 m n 451 18 m ( 12 m + 6 n ) 16 8 log 16 8 + ( 8 m + 4 n ) 20 9 log 20 9 + ( 12 m n 2 m 4 n ) 25 10 log 25 10 + ( 24 m n + 4 m 4 n ) 35 12 log 35 12 + ( 6 m n 21 m + 21 n ) 49 14 log 49 14
= l o g ( ( 1309 18 ) n + 121 m n ( 451 18 ) m ) ( 53.34291666 m + 85.84556667 n + 128.7255000 m n ( 1309 18 ) n + 121 m n ( 451 18 ) m )
14.
N E N T N R e Z G 3 ( Γ ) = log ( 17196 m n 10250 m + 13214 n ) 1 17196 m n 10250 m + 13214 n [ ( 12 m + 6 n ) ( 128 ) log 128 + ( 8 m + 4 n ) ( 180 ) log 180 + ( 12 m n 2 m 4 n ) ( 250 ) log 250 + ( 24 m n + 4 m 4 n ) ( 420 ) log 420 + ( 6 m n 21 m + 21 n ) ( 686 ) log 686 ]
= l o g ( 17196 m n 10250 m + 13214 n ) ( 71766.5994 m + 85880.2374 n + 1.043319084 × 10 5 m n 17196 m n 10250 m + 13214 n )
where N A B C ( Γ ) = 3 2 3 m + ( 3 2 ) 2 3 n + ( 4 5 ) 7 5 m + ( 2 5 ) 7 5 n + ( 24 5 ) 2 m n ( 4 5 ) 2 m ( 8 5 ) 2 n + ( 24 7 ) 2 7 m n + ( 4 7 ) 2 7 m ( 4 7 ) 2 7 n + ( 12 7 ) 3 m n 6 3 m + 6 3 n and χ ( Γ ) = 3 2 m + ( 3 2 ) 2 n + ( 8 3 ) m + ( 4 3 ) n + ( 6 5 ) 5 2 m n ( 1 5 ) 5 2 m ( 2 5 ) 5 2 n + 4 3 m n + ( 2 3 ) 3 m ( 2 3 ) 3 n + ( 3 7 ) 2 7 m n ( 3 2 ) 2 7 m + ( 3 2 ) 2 7 n .
The closed-form expressions of the neighbourhood degree sum-based indices computed in Theorem 5 and Theorem 6 are represented as 3D plots in this section. These graphical representations provide a visual interpretation of the mathematical expressions to study the indices with respect to the molecular structure. In addition, the results of Theorem 2 have also been represented as comparison plots to understand the behaviour of the indices with respect to each other. These plots are presented in Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.
Figure 8. 3D plots for theorem 5.
Figure 8. 3D plots for theorem 5.
Molecules 28 00168 g008aMolecules 28 00168 g008b
Figure 9. 3D plots for Table 8.
Figure 9. 3D plots for Table 8.
Molecules 28 00168 g009
Figure 10. 3D plots for Table 9.
Figure 10. 3D plots for Table 9.
Molecules 28 00168 g010
Figure 11. 3D plots for Table 10.
Figure 11. 3D plots for Table 10.
Molecules 28 00168 g011
Figure 12. 3D plots for Table 11.
Figure 12. 3D plots for Table 11.
Molecules 28 00168 g012

9. Neighbourhood Degree Sum-Based M-polynomial and Entropy Measures for β-Graphdiyne

Graphdiyne, a recently developed two-dimensional carbon allotrope, has been gaining prominence. This carbon allotrope has gained increasing interest due to its intriguing electronic properties, including high mobility and conductivity, good field emission properties, and a tunable natural band gap, in addition to its distinctive porous structure. It is predicted to have good application prospects in a variety of fields, including gas separation, catalysis, water remediation, humidity sensor, and energy-related fields [33]. It consists of V ( Γ ) = 60 m n + 2 m + 34 n and E ( Γ ) = 66 n m + 35 m + n . β-Graphdiyne is represented in Figure 13. The edge partitions of β-Graphdiyne are presented in Table 12.
Recently, Julietraja and Venugopal [34] studied degree-based indices using M-Polynomials for a coronoid system. Chu et al. [35] studied degree-and irregularity-based indices for benzenoids. Julietraja et al. [36] investigated degree-based indices for certain classes of benzenoid systems. Julietraja et al. [37] investigated degree-based indices for a donut-benzenoid system. Julietraja et al. [38] investigated irregularity-based indices for PAHs. Liu et al. [39] studied the network coherence analysis on a family of nested weighted N-Polygon networks. Liu et al. [40] studied the analyses of some structural properties on a class of hierarchical scale-free networks. Liu et al. [41] studied minimizing the Kirchhoff index among graphs with a given vertex bipartiteness. Liu et al. [42] studied the Zagreb indices and multiplicative Zagreb indices of Eulerian Graphs. Liu et al. [43] studied the valency-based topological descriptors and structural properties of the generalized Sierpinski networks. Liu et al. [44] studied the Kirchhoff index and spanning trees of the Möbius/cylinder octagonal chain.
Theorem 7.
If Γ is a β- graphdiyne system, then N M -polynomial of Γ is given as follows
N M ( Γ ; 𝓇 , 𝓉 ) = ( 12 m n + 26 m + 10 n ) 𝓇 4 𝓉 4 + ( 24 m n + 4 m 4 n ) 𝓇 4 𝓉 5                                                       + ( 24 m n + 4 m 4 n ) 𝓇 5 𝓉 7 + ( 6 m n + m n ) 𝓇 7 𝓉 7
Proof. 
Let f ( 𝓇 , 𝓉 ) = N M ( Γ ; 𝓇 , 𝓉 ) = ( 12 m n + 26 m + 10 n ) 𝓇 4 𝓉 4 + ( 24 m n + 4 m 4 n ) 𝓇 4 𝓉 5 + ( 24 m n + 4 m 4 n ) 𝓇 5 𝓉 7 + ( 6 m n + m n ) 𝓇 7 𝓉 7
Theorem 8.
If Γ is a β- graphdiyne system, then N M -entropy measures of Γ are given as follows
1.
N M 1 ( Γ ) = 684 m n + 306 m 18 n
2.
N M 2 ( Γ ) = 1806 m n + 685 m 109 n
3.
N M 2 m ( Γ ) = ( 2703 980 ) m n + ( 3841 1960 ) m + ( 569 1960 ) n
4.
N R α ( Γ ) = 16 φ ( 12 m n + 26 m + 10 n ) + 20 φ ( 24 m n + 4 m 4 n ) + 35 φ ( 24 m n + 4 m 4 n ) + 49 φ   ( 6 m n + m n )
5.
N R R α ( Γ ) = 1 / 16 φ   ( 12 m n + 26 m + 10 n ) + 1 / 20 φ   ( 24 m n + 4 m 4 n ) + 1 / 35 φ ( 24 m n + 4 m 4 n ) + 1 / 49 φ   ( 6 m n + m n )
6.
N S D D ( Γ ) = ( 4758 35 ) m n + ( 2473 35 ) m + ( 47 35 ) n
7.
N H ( Γ ) = ( 277 21 ) m n + ( 1033 126 ) m + ( 101 126 ) n
8.
N I ( Γ ) = ( 505 3 ) m n + ( 1369 18 ) m ( 73 18 ) n
9.
N A ( Γ ) = ( 219777191 98784 ) m n + ( 489523367 592704 ) m ( 84904103 592704 ) n
10.
N A B C ( Γ ) = 3 3 2 m n + 13 2 3 2 m + 5 2 3 2 n + 12 5 7 5 m n + 2 5 7 5 m 2 5 7 5 n + 24 7 2 7 m n + 4 7 2 7 m 4 7 2 7 n + 12 7 3 m n + 2 7 3 m 2 7 3 n
11.
N G A ( Γ ) = 18 m n + 27 m + 9 n + ( 32 3 )   5 m n + ( 16 9 )   5 m ( 16 9 )   5 n + 4 7   5 m n + ( 2 3 ) 7   5 m ( 2 3 ) 7   5 n
12.
N B 1 ( Γ ) = 1788 m n + 778 m 58 n
13.
N B 2 ( Γ ) = 5016 m n + 1988 m 260 n
14.
N H B 1 ( Γ ) = 324312 m n + 81700 m 40228 n

10. Neighbourhood Degree Sum-Based Entropy Measures for β-Graphdiyne

Theorem 9.
If Γ is a β- graphdiyne system, then N M -entropy measures of Γ are given as follows
1.
N E N T N M 1 ( Γ ) = log ( 684 m n + 306 m 18 n ) 1 684 m n + 306 m 18 n [ ( 12 m n + 26 m + 10 n ) ( 8 ) log 8 + ( 24 m n + 4 m 4 n ) ( 9 ) log 9 + ( 24 m n + 4 m 4 n ) ( 12 ) log 12 + ( 6 m n + m n ) ( 14 ) log 14 ]
= log ( 684 m n + 306 m 18 n ) ( 1611.5532 m n + 667.8370 m 68.9698 n 684 m n + 306 m 18 n )
2.
N E N T N M 2 ( Γ ) = log ( 1806 m n + 685 m 109 n ) 1 1806 m n + 685 m 109 n [ ( 12 m n + 26 m + 10 n ) ( 16 ) log 16 + ( 24 m n + 4 m 4 n ) ( 20 ) log 20 + ( 24 m n + 4 m 4 n ) ( 35 ) log 35 + ( 6 m n + m n ) ( 49 ) log 49 ]
= l o g ( 1806 m n + 685 m 109 n ) ( 6100.9164 m n + 2081.4978 m 484.4802 n 1806 m n + 685 m 109 n )
3.
N E N T N M 2 m ( Γ ) = log 2703 980 m n + 3841 1960 m + 569 1960 n 1 2703 980 m n + 3841 1960 m + 569 1960 n ( 12 m n + 26 m + 10 n ) 1 16 log 1 16 + ( 24 m n + 4 m 4 n ) 1 20 log 1 20 + ( 24 m n + 4 m 4 n ) 1 35 log 1 35 + ( 6 m n + m n ) 1 49 log 1 49
= l o g ( ( 2703 980 ) m n + ( 3841 1960 ) m + ( 569 1960 ) n ) ( 8.588756939 m n 5.590359490 m .6479905100 n ( 2703 980 ) m n + ( 3841 1960 ) m + ( 569 1960 ) n )
4.
N E N T N H M ( Γ ) = log ( 7344 m n + 2760 m 456 n ) 1 7344 m n + 2760 m 456 n [ ( 12 m n + 26 m + 10 n ) ( 64 ) log 64 + ( 24 m n + 4 m 4 n ) ( 81 ) log 81 + ( 24 m n + 4 m 4 n ) ( 144 ) log 144 + ( 6 m n + m n ) ( 196 ) log 196 ]
= log ( 7344 m n + 2760 m 456 n ) 35119.5408 m n + 12241.3272 m 2659.2216 n 7344 m n + 2760 m 456 n
5.
N E N T N A ( Γ ) = log 219777191 98784 m n + 489523367 592704 m 84904103 592704 n 1 219777191 98784 m n + 489523367 592704 m 84904103 592704 n [ ( 12 m n + 26 m + 10 n ) 16 6 3 log 16 3 6 + ( 24 m n + 4 m 4 n ) 20 7 3 log 20 3 7 + ( 24 m n + 4 m 4 n ) 35 10 3 log 35 2 10 + ( 6 m n + m n ) 49 12 3 log 49 3 12
= l o g ( ( 219777191 98784 ) m n + ( 489523367 592704 ) m ( 84904103 592704 ) n ) ( 4023.297013 m n + 1099.996898 m 455.8258044 n ( 219777191 98784 ) m n + ( 489523367 592704 ) m ( 84904103 592704 ) n )
6.
N E N T N A B C ( Γ ) = log ( N A B C ( Γ ) ) 1 N A B C ( Γ ) ( 12 m n + 26 m + 10 n ) 6 16 log 6 16 + ( 24 m n + 4 m 4 n ) 7 20 log 7 20 + ( 24 m n + 4 m 4 n ) 10 35 log 10 35 + ( 6 m n + m n ) 12 49 log 12 49
7.
N E N T N G A ( Γ ) = log ( N G A ( Γ ) ) 1 N G A ( Γ ) ( 12 m n + 26 m + 10 n ) ( 1 ) log 1 + ( 24 m n + 4 m 4 n ) 2 20 9 log 2 20 9 + ( 24 m n + 4 m 4 n ) 2 35 12 log 2 35 12 + ( 6 m n + m n ) ( 1 ) log 1
8.
N E N T N A G 1 ( Γ ) = log N A G 1 ( Γ ) 1 N A G 1 ( Γ ) ( 12 m n + 26 m + 10 n ) ( 1 ) log 1 + ( 24 m n + 4 m 4 n ) 9 2 20 log 9 2 20 + ( 24 m n + 4 m 4 n ) 12 2 35 log 12 2 35 + ( 6 m n + m n ) ( 1 ) log 1
9.
N E N T N F 1 ( Γ ) = log ( 3732 m n + 1390 m 238 n ) 1 3732 m n + 1390 m 238 n [ ( 12 m n + 26 m + 10 n ) ( 32 ) log 32 + ( 24 m n + 4 m 4 n ) ( 41 ) log 41 + ( 24 m n + 4 m 4 n ) ( 74 ) log 74 + ( 6 m n + m n ) ( 98 ) log 98 ]
= l o g ( 3732 m n + 1390 m 238 n ) ( 15325.1112 m n + 5215.9196 m 1223.3180 n 3732 m n + 1390 m 238 n )
10.
N E N T N F 2 ( Γ ) = log ( 54318 m n + 15197 m 5981 n ) 1 54318 m n + 15197 m 5981 n [ ( 12 m n + 26 m + 10 n ) ( 256 ) log 256 + ( 24 m n + 4 m 4 n ) ( 400 ) log 400 + ( 24 m n + 4 m 4 n ) ( 1225 ) log 1225 + ( 6 m n + m n ) ( 2401 ) log 2401 ]
= l o g ( 54318 m n + 15197 m 5981 n ) ( 3.957344760 × 10 5 m n + 1.000254548 × 10 5 m 48920.8916 n 54318 m n + 15197 m 5981 n )
11.
N E N T N χ ( Γ ) = log ( χ ( Γ ) ) 1 χ ( Γ ) ( 12 m n + 26 m + 10 n ) 1 8 log 1 8 + ( 24 m n + 4 m 4 n ) 1 3 log 1 3 + ( 24 m n + 4 m 4 n ) 1 12 log 1 12 + ( 6 m n + m n ) 1 14 log 1 14
12.
N E N T N R e Z Γ 1 ( Γ ) = log 936 35 m n + 576 35 m + 54 35 n 1 936 35 m n + 576 35 m + 54 35 n ( 12 m n + 26 m + 10 n ) 1 2 log 1 2 + ( 24 m n + 4 m 4 n ) 9 20 log 9 20 + ( 24 m n + 4 m 4 n ) 12 35 log 12 35 + ( 6 m n + m n ) 2 7 log 2 7
= l o g ( ( 936 35 ) m n + ( 576 35 ) m + ( 54 35 ) n ) ( 23.73832800 m n 12.27418800 m .202512000 n ( 936 35 ) m n + ( 576 35 ) m + ( 54 35 ) n )
13.
N E N T N R e Z G 2 ( Γ ) = log 505 3 m n + 1369 18 m 73 18 n 1 505 3 m n + 1369 18 m 73 18 n [ ( 12 m n + 26 m + 10 n ) ( 2 ) log 2 + ( 24 m n + 4 m 4 n ) 20 9 log 20 9 + ( 24 m n + 4 m 4 n ) 35 12 log 35 12 + ( 6 m n + m n ) 7 2 log 7 2
= l o g ( ( 505 3 ) m n + ( 1369 18 ) m ( 73 18 ) n ) ( 160.4596000 m n + 60.01446667 m 10.10766667 n ( 505 3 ) m n + ( 1369 18 ) m ( 73 18 ) n )
14.
N E N T N R e Z G 3 ( Γ ) = log ( 20052 m n + 6414 m 1806 n ) 1 20052 m n + 6414 m 1806 n [ ( 12 m n + 26 m + 10 n ) ( 128 ) log 128 + ( 24 m n + 4 m 4 n ) ( 180 ) log 180 + ( 24 m n + 4 m 4 n ) ( 420 ) log 420 + ( 6 m n + m n ) ( 686 ) log 686 ]
= l o g ( 20052 m n + 6414 m 1806 n ) ( 1.176538404 × 10 5 m n + 34514.3174 m 12156.3014 n 20052 m n + 6414 m 1806 n )
where N A B C ( Γ ) = 3 3 2 m n + ( 13 2 ) 3 2 m + ( 5 2 ) 3 2 n + 49 14 ( 12 5 ) 7 5 m n + ( 2 5 ) 7 5 m ( 2 5 ) 7 5 n + ( 24 7 ) 2 7 m n + ( 4 7 ) 2 7 m ( 4 7 ) 2 7 n + ( 12 7 ) 3 m n + ( 2 7 ) 3 m ( 2 7 ) 3 n , N G A ( Γ ) = 18 m n + 27 m + 9 n + ( 32 3 ) 5 m n + ( 16 9 ) 5 m ( 16 9 ) 5 n + 4 7 5 m n + ( 2 3 ) 7 5 m ( 2 3 ) 7 5 n , N A G 1 = 18 m n + 27 m + 9 n + ( 54 5 ) 5 m n + ( 9 5 ) 5 m ( 9 5 ) 5 n + ( 144 35 ) 7 5 m n + ( 24 35 ) 7 5 m ( 24 35 ) 7 5 n and N χ ( Γ ) = 3 2 m n + ( 13 2 ) 2 m + ( 5 2 ) 2 n + 8 m n + ( 4 3 ) m ( 4 3 ) n + 4 3 m n + ( 2 3 ) 3 m ( 2 3 ) 3 n + ( 3 7 ) 2 7 m n + ( 1 14 ) 2 7 m ( 1 14 ) 2 7 n .
In this section, the neighbourhood degree sum-based indices computed in Theorem 8 and Theorem 9 are plotted as 3D graphs to provide a visual tool for understanding the behavioural pattern of the indices for changing molecular structures. In addition, the results of Theorem 8 have also been presented as comparison plots in Figure 14 to study the behaviour of the indices with respect to each other and the molecular structure. The numerical values of the indices are computed and listed in Table 13, Table 14, Table 15 and Table 16. The numerical values are represented as 3D plots in Figure 15, Figure 16, Figure 17, and Figure 18 respectively to provide a visual understanding to the readers.

11. Conclusions

In this article, the closed-form analytical expressions of neighbourhood sum degree-based indices of three types of graphenes have been derived using the M-Polynomial. The computed indices are presented as individual 3D plots and comparison plots for a convenient interpretation of the mathematical expressions. The neighbourhood sum degree-based entropy measures have also been calculated for the three types of graphene structures. These indices are also visualised as 3D plots to corroborate the dependence of the indices on the underlying molecular structure. These indices have not been studied before for these structures; hence, this study is one of a kind. This study will enable future researchers to explore more topological indices for these fascinating structures.

Author Contributions

Conceptualization, J.K. and A.A.R.S.; methodology, J.K.; software, J.K.; validation, J.K., and A.A.R.S.; investigation, A.A.R.S. and J.K.; writing—original draft preparation, J.K.; writing—review and editing, J.K.; visualization, J.Y.; supervision, J.Y.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript.

Funding

The Authors would like to thank the “Doctoral research funds of Chaohu University 2023”.

Institutional Review Board Statement

Not Applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No data associated in the manuscript.

Acknowledgments

The Authors would like to thank the “Doctoral research funds of Chaohu University 2023”.

Conflicts of Interest

The authors declare no conflict of interest.

Sample Availability

Not applicable.

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Figure 1. β -Graphene (7,7).
Figure 1. β -Graphene (7,7).
Molecules 28 00168 g001
Figure 2. 3D plots for Theorem 2.
Figure 2. 3D plots for Theorem 2.
Molecules 28 00168 g002
Figure 3. 3D Plots for Table 3.
Figure 3. 3D Plots for Table 3.
Molecules 28 00168 g003
Figure 4. 3D Plots for Table 4.
Figure 4. 3D Plots for Table 4.
Molecules 28 00168 g004
Figure 5. 3D Plots for Table 5.
Figure 5. 3D Plots for Table 5.
Molecules 28 00168 g005
Figure 6. 3D Plots for Table 6.
Figure 6. 3D Plots for Table 6.
Molecules 28 00168 g006
Figure 7. β -Graphyne (7,7).
Figure 7. β -Graphyne (7,7).
Molecules 28 00168 g007
Figure 13. β -Graphdiyne (7,7).
Figure 13. β -Graphdiyne (7,7).
Molecules 28 00168 g013
Figure 14. 3D plots for theorem 8.
Figure 14. 3D plots for theorem 8.
Molecules 28 00168 g014
Figure 15. 3D plots for Table 13.
Figure 15. 3D plots for Table 13.
Molecules 28 00168 g015
Figure 16. 3D plots for Table 14.
Figure 16. 3D plots for Table 14.
Molecules 28 00168 g016
Figure 17. 3D plots for Table 15.
Figure 17. 3D plots for Table 15.
Molecules 28 00168 g017
Figure 18. 3D plots for Table 16.
Figure 18. 3D plots for Table 16.
Molecules 28 00168 g018
Table 1. N M -Polynomial Expressions.
Table 1. N M -Polynomial Expressions.
S. NoDegree-Based TIs Derived from N M ( Γ ; 𝓇 , 𝓽 )
1. M 1 ( Γ ) = D 𝓇 + D 𝓽 ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 𝓽 = 1
2. M 2 ( Γ ) = D 𝓇 D 𝓽 ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 𝓽 = 1
3. M 2 m ( Γ ) = S 𝓇 S 𝓽 ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 𝓽 = 1
4. R α ( Γ ) = D 𝓇 α D 𝓽 α ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 𝓽 = 1
5. R R α ( Γ ) = S 𝓇 α S 𝓽 α ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 𝓽 = 1
6. S D D ( Γ ) = ( D 𝓇 S 𝓽 + D 𝓽 S 𝓇 ) ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 𝓽 = 1
7. H ( Γ ) = 2 S 𝓇 J ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 1
8. I ( Γ ) = S 𝓇 J D 𝓇 D 𝓽 ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 1
9. A ( Γ ) = S 𝓇 3 Q 2   J D 𝓇 3 D 𝓽 3 ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 1
10. A B C ( Γ ) = D 𝓇 1 2   Q 2 J S 𝓇 1 2 S 𝓽 1 2 ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 1
11. G A ( Γ ) = 2 S 𝓇 J D 𝓇 1 2 D 𝓽 1 2 ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 1
12. B 1 ( Γ ) = ( D 𝓇 + D 𝓽 + 2 D 𝓇 Q 2 J ) ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 𝓽 = 1
13. B 2 ( Γ ) = D 𝓇 Q 2 J ( D 𝓇 + D 𝓽 ) ( N M ( Γ ; 𝓇 , 𝓽 ) ) | 𝓇 = 1
14. H B 1 ( Γ ) = D 𝓇 2 + D 𝓽 2 + 2 D 𝓇 2 Q 2 J + 2 D 𝓇 Q 2 J D 𝓇 + D 𝓽 ( N M ( Γ ; 𝓇 , 𝓽 ) ) 𝓇 = 𝓽 = 1
Table 2. Partition table for β -Graphene.
Table 2. Partition table for β -Graphene.
Edge TypesFrequency
d 55 4 m + 2 n
d 57 4 m + 8
d 58 4 m + 4 n 8
d 79 2 m + 4
d 88 2 n 2
d 89 8 m + 4 n 12
d 99 18 m n 21 m n + 10
Table 3. Comparison Table for Theorem 2.
Table 3. Comparison Table for Theorem 2.
m = n N M M 1 ( Γ ) N M M 2 ( Γ ) N M M 2 m ( Γ ) N M R α ( Γ ) N M R R α ( Γ ) N M S D D ( Γ ) N M H ( Γ )
140814340.75202.614.6561.754.62
2146457541.93728.5813.26196.3013.18
33168129903.551578.5325.88402.8525.75
45520231425.622752.4942.49681.4042.31
58520362108.134250.4563.101032.0062.88
Table 4. Comparison Table for Theorem 2.
Table 4. Comparison Table for Theorem 2.
m = n N M I ( Γ ) N M A ( Γ ) N M A B C ( Γ ) N M G A ( Γ ) N M B 1 ( Γ ) N M B 2 ( Γ ) N M H B 1 ( Γ )
1100.622022.8315.3029.7911044992509712
2362.618614.4246.6195.484008202682464404
3786.5919876.8793.91197.178712459125912088
41372.5735810.19157.22334.86152168192410852764
52120.5556414.37236.53508.562352012830417286432
Table 5. Comparison Table for Theorem 3.
Table 5. Comparison Table for Theorem 3.
m = n N E N T N M 1 ( Γ ) N E N T N M 2 ( Γ ) N E N T N M 2 m ( Γ ) N E N T N H M ( Γ ) N E N T N A ( Γ ) N E N T N A B C ( Γ ) N E N T N G A ( Γ )
13.383.313.323.314.423.403.40
24.544.494.464.495.344.564.56
35.275.235.185.236.005.285.29
45.805.775.725.776.495.815.82
56.226.206.156.206.896.236.23
Table 6. Comparison Table for Theorem 3.
Table 6. Comparison Table for Theorem 3.
m = n N E N T N A G 1 ( Γ ) N E N T N F 1 ( Γ ) N E N T N F 2 ( Γ ) N E N T N χ ( Γ ) N E N T N R E Z G 1 ( Γ ) N E N T N R E Z G 2 ( Γ ) N E N T N R E Z G 3 ( Γ )
13.403.313.086.803.694.393.21
24.564.504.358.414.825.524.42
35.295.245.139.355.536.225.18
45.825.775.7010.006.056.745.73
56.236.206.1310.516.477.166.17
Table 7. Partition Table for β - Graphyne.
Table 7. Partition Table for β - Graphyne.
Edge TypesFrequency
d 44 12 m + 6 n
d 45 8 m + 4 n
d 55 12 m n 2 m 4 n
d 57 24 m n + 4 m 4 n
d 77 6 m n 21 m + 21 n
Table 8. Comparison Table for Theorem 5.
Table 8. Comparison Table for Theorem 5.
m = n N M M 1 ( Γ ) N M M 2 ( Γ ) N M M 2 m ( Γ ) N M R α ( Γ ) N M R R α ( Γ ) N M S D D ( Γ ) N M H ( Γ )
168418122.77339.6513.30135.3413.22
2235264928.121167.2741.22444.1740.96
350041404016.052482.8783.78926.4983.21
486402445626.554286.44140.951582.29139.98
5132603774039.636577.98212.772411.57211.26
Table 9. Comparison Table for Theorem 5.
Table 9. Comparison Table for Theorem 5.
m = n N M I ( Γ ) N M A ( Γ ) N M A B C ( Γ ) N M G A ( Γ ) N M B 1 ( Γ ) N M B 2 ( Γ ) N M H B 1 ( Γ )
1168.672241.8337.3265.5917885988326316
2579.338091.08119.80214.516192216721253976
31232.0017547.76247.46446.7713212470522782980
42126.6730611.88420.29762.3422848821284913328
53263.3347283.42638.291161.24351001269007645020
Table 10. Comparison Table for Theorem 6.
Table 10. Comparison Table for Theorem 6.
m = n N E N T M 1 ( Γ ) N E N T M 2 ( Γ ) N E N T M 2 m ( Γ ) N E N T H M ( Γ )   N E N T A ( Γ ) N E N T A B C ( Γ ) N E N T G A ( Γ )
14.174.124.122.615.904.194.19
25.365.325.314.527.085.375.38
36.106.066.055.527.816.116.11
46.636.606.596.198.346.646.64
57.057.037.026.708.777.067.06
Table 11. Comparison Table for Theorem 6.
Table 11. Comparison Table for Theorem 6.
m = n N E N T A G 1 ( Γ ) N E N T F 1 ( Γ ) N E N T F 2 ( Γ ) N E N T N χ ( Γ ) N E N T R E Z G 1 ( Γ ) N E N T R E Z G 2 ( Γ ) N E N T R E Z G 3 ( Γ )
14.194.333.624.194.174.174.04
25.385.554.855.375.365.365.26
36.116.305.606.116.106.106.01
46.646.856.156.646.636.636.55
57.067.286.587.067.057.056.98
Table 12. Partition Table for β -Graphdiyne.
Table 12. Partition Table for β -Graphdiyne.
Edge TypesFrequency
d 44 12 m n + 26 m + 10 m
d 45 24 m n + 4 m 4 n
d 57 24 m n + 4 m 4 n
d 77 6 m n + m n
Table 13. Comparison Table for Theorem 8.
Table 13. Comparison Table for Theorem 8.
m = n N M M 1 ( Γ ) N M M 2 ( Γ ) N M M 2 m ( Γ ) N M R α ( Γ ) N M R R α ( Γ ) N M S D D ( Γ ) N M H ( Γ )
197223825.01483.3222.28207.9422.19
23312837615.531645.2771.12687.7770.76
370201798231.573485.85146.521439.49145.71
4120963120053.136005.07248.482463.09247.05
5185404803080.209202.93377.013758.57374.76
Table 14. Comparison Table for Theorem 8.
Table 14. Comparison Table for Theorem 8.
m = n N M I ( Γ ) N M A ( Γ ) N M A B C ( Γ ) N M G A ( Γ ) N M B 1 ( Γ ) N M B 2 ( Γ ) N M H B 1 ( Γ )
1240.332907.4959.39101.5225086744365784
2817.3310264.64193.47334.078592235201380192
31731.0022071.43402.24697.6618252503283043224
42981.3338327.88685.721192.2531488871685354880
54568.3359033.981043.851817.89483001340408315160
Table 15. Comparison Table for Theorem 9.
Table 15. Comparison Table for Theorem 9.
m = n N E N T M 1 ( Γ ) N E N T M 2 ( Γ ) N E N T M 2 m ( Γ ) N E N T H M ( Γ )   N E N T A ( Γ ) N E N T A B C ( Γ ) N E N T G A ( Γ )
14.614.544.574.546.374.624.63
25.805.745.765.747.545.815.82
36.536.486.496.478.276.556.55
47.077.017.037.018.817.097.09
57.497.447.457.449.237.517.51
Table 16. Comparison Table for Theorem 9.
Table 16. Comparison Table for Theorem 9.
m = n N E N T A G 1 ( Γ ) N E N T F 1 ( Γ ) N E N T F 2 ( Γ ) N E N T χ ( Γ ) N E N T R E Z G 1 ( Γ ) N E N T R E Z G 2 ( Γ ) N E N T R E Z G 3 ( Γ )
14.634.544.034.624.614.614.44
25.825.735.225.815.805.805.64
36.556.475.966.556.546.546.38
47.097.016.507.097.077.076.92
57.517.436.937.517.507.497.35
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Yang, J.; Konsalraj, J.; Raja S., A.A. Neighbourhood Sum Degree-Based Indices and Entropy Measures for Certain Family of Graphene Molecules. Molecules 2023, 28, 168. https://doi.org/10.3390/molecules28010168

AMA Style

Yang J, Konsalraj J, Raja S. AA. Neighbourhood Sum Degree-Based Indices and Entropy Measures for Certain Family of Graphene Molecules. Molecules. 2023; 28(1):168. https://doi.org/10.3390/molecules28010168

Chicago/Turabian Style

Yang, Jun, Julietraja Konsalraj, and Arul Amirtha Raja S. 2023. "Neighbourhood Sum Degree-Based Indices and Entropy Measures for Certain Family of Graphene Molecules" Molecules 28, no. 1: 168. https://doi.org/10.3390/molecules28010168

APA Style

Yang, J., Konsalraj, J., & Raja S., A. A. (2023). Neighbourhood Sum Degree-Based Indices and Entropy Measures for Certain Family of Graphene Molecules. Molecules, 28(1), 168. https://doi.org/10.3390/molecules28010168

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