Degree-based topological indices and entropies of diamond crystals

Abstract

High hardness, low friction coefficient and chemical resistance are only a few of the exceptional mechanical qualities of diamond. Diamonds can be artificially created to have different levels of conductivity, or they can be single, micro or nanocrystalline and highly electrically insulating. It also has high biocompatibility and is famous for being mechanically robust. Due to its high hardness, lack of ductility and difficulty in welding, diamond is a challenging material to construct devices with. Diamonds have experienced a rise in attention as a biological material in recent decades due to new synthesis and fabrication techniques that have eliminated some of these disadvantages. In general, entropic measurements are used for investigating the chemical or biological properties of molecular structures. This study calculates several important $K$ -Banhatti entropies, redefined Zagreb entropies and atom-bond sum connectivity entropy for diamond crystals. We also present a numeric and graphical explanations of obtain indices.

Introduction

Diamond is regarded as a suitable strengthening stage of metal matrix composites because it possesses the maximum heat conductivity of any natural substance, a low rate of thermal expansion, extreme durability and exceptional chemical solidity.^1,2 The artificial diamond market has grown recently, and this has culminated in a large drop in price, which encourages the use of artificial diamond in composite materials. ²

Due to its outstanding properties, diamond is a valued material for a wide range of uses. For instance, this carbon allotrope's tetrahedral geometry with covalent carbon bonds allows for the greatest thermal conductivity, the lowest heat expansion coefficient (0.8 × 10−6 K-1), and exceptionally high hardness (98 GPa).^1,3–6

Additionally, diamond is biocompatible and has a strong chemical resistance to bases and acids. ⁷ Carbon is one of the lightest elements and makes up diamonds as well. An isotope effect's size is directly related to the square root of the mass ratio among the different isotopes. Where this ratio is biggest, the lighter elements are where isotope effects are most prominent. The stable isotopes of carbon are ¹²C (98.9 percentage) and ¹³C (1.1 percentage), which occur naturally.

Simple cryogenic distillation can be used to separate carbon isotopes due to the comparatively large percentage variation in mass. This has led to the commercial availability of a wide range of carbon compounds with high isotopic purity. ⁸ The lattice of tightly coupled, extremely symmetric crystals formed by covalent bonds is formed by the lowest mass element, which is diamond. The material is currently attracting enormous interest for a variety of optical applications due to its exceptional qualities that arise from recent advancements in its synthesis.

There is a tantalizing possibility of much increased capability because the optical properties are distinct from those of other materials in many ways. The assistance of optical design greatly depends on having a thorough understanding of how electromagnetic radiation interacts with diamond's bulk and surface. ⁹ The results of the simulation showed that too much oxygen increases the oxidation of diamond, which raises the substance's mechanical rigidity and roughness rate. The lower rigidity of the oxidized diamond allowed for a larger surface deformation.^10–12

Due to its advantageous characteristics, such as its malleability and mechanical workability, the alloy of copper is widely used in diamond instruments. ¹³ Copper is a typical bond material used for numerous diamond-cutting instruments due to its inexpensive cost and low sintering temperature. ¹⁴ The diamond swords can be properly squeezed and self-sharpen when a suitable ratio of diamonds is established and enough cutting edges are scattered across the working surface of the diamond tool. In addition, the cutting life and efficiency of diamond particles is directly affected by their size.^15–17

Several methods exist for converting a chemical network into a graph. Examine the types of compounds that are present in the network. ¹⁸ These may involve other chemicals, ions or radicals. ¹⁹ Find out which species in the network are involved in which chemical processes. This can entail looking over experimental results, chemical equations or other knowledge sources. ²⁰

Al-Raee ²¹ initiated the formula for calculating the specific bond volume of the Morse potential is universal and can be used for many materials. Shi et al. studied the stability of the interface is determined by the type of interfacial border and bond, as well as the populations and numbers of interfacial bonds. ²² The layer-projected density of states analysis indicates that all the examined contacts display metallic properties. ²² Zhang et al. presents the prediction of a new carbon phase, called pentapeptide diamond, which exhibits orthorhombic crystal structure and sp³ bonding. ²³ The prediction was made by integrating the particle swarm optimization method with first-principle calculations. The phonon spectra, total energy and elastic constants calculations of the pentapeptide diamond provide evidence that it is dynamically, thermally and mechanically stable under zero pressure. It has a large bulk modulus of 385 GPa and a Vickers hardness of 72.6 GPa, which is like that of diamond. The electronic band structure simulations indicate that the pentapeptide diamond possesses a direct band gap of 4.18 electron volts (eV). ²³

Make an edge connecting the points that represent the reactants and outcomes of each chemical reaction in the graph. ²⁴ Complete the graph by adding any other details, such as rates of reactions or thermodynamic characteristics, that might be essential to the chemical network.^25,26 Once the chemical network is represented as a graph, its composition can be examined using a range of graph-theoretical methods and procedures.^27–29 For instance, strongly connected node groups can be detected using community detection techniques, and important nodes in the network can be located using centrality metrics.^30,31

This method can offer guidance for the development of novel chemical processes and reactions as well as knowledge about the chemistry of intricate chemical structures. ³² Topological indices play an essential role in providing guidance for the treatment of tumours or malignancies.^33,34 Experimental or numerical methods can be used to find these indices. Computer analysis offers a time- and money-efficient solution because experimental data, despite their high cost, are important. The eccentricity-based topological indices are a key component of the theory of chemical graphs. ³⁵ Chemist Wiener created the first topological index in 1947. ³⁶

Methodology

Consider a graph with the form $G=(V,E)$ with the vertex position $V=V(G)$ and the edge group $E=E(G)$ . The number of edges that join a vertex $C_i$ defines its degree, and we represent it by the notation ${\alpha }_{C_i}$ . For any edge $e=C_1\sim C_2$ of G, the degree of e is defined as ${\alpha }_e={\alpha }_{C_1}+{\alpha }_{C_2}-2$ . A molecular graph, often mention to as a chemical network in graph theory, is a straightforward network with vertices and edges that represent individual atoms and chemical bonds. ³⁷ We use eight different degree-based topological indices to find the entropy in this paper. The formula of each of index is presented in Table 1 and they are introduced as follows: four Banhatti indices were first computed by Kulli in 2016;^38,39 concepts of atom-bond and sum connectivity indices were collected by Ali et al., which led to the create another molecular descriptor known as the atom-bond sum-connectivity index in. ⁴⁰ We use the method of edge partition to compute our results. By observing the degrees of vertices and edges of the underline structure (graph) G, we partition the edge group $E(G)$ into parts of the form:

$$E^{(x,y)}=\{e=C_1\sim C_2\in E(G)|x={\alpha }_{C_1},y={\alpha }_{C_2}\}.$$

Table 1.

Degree-based topological indices.

Index	Symbol	Formula
$1^{st}$ redefined Zagreb	$ReZ{G}_1$	$\sum _{{\mathrm{C}}_1\sim C_2}\frac{{\alpha }_{C_1}+{\alpha }_{C_2}}{{\alpha }_{C_1}{\times }_{{\alpha }_{C_2}}}$
$2^{nd}$ redefined Zagreb	$ReZ{G}_2$	$\sum _{{\mathrm{C}}_1\sim C_2}\frac{{\alpha }_{C_1}\times {\alpha }_{C_2}}{{\alpha }_{C_1}+{\alpha }_{C_2}}$
$3^{rd}$ redefined Zagreb	$ReZ{G}_3$	$\sum _{C_1\sim C_2}({\alpha }_{C_1}\times {\alpha }_{C_2})({\alpha }_{C_1}+{\alpha }_{C_2})$
Atom-bond sum connectivity	$ABS$	$\sum _{C_1\sim C_2}\sqrt{\frac{({\alpha }_{C_1}+{\alpha }_{C_2}-2)}{({\alpha }_{C_1}+{\alpha }_{C_2})}}$
$1^{st}$ $K$ -Banhatti	$B_1$	$\sum _{e=C_1\sim C_2}({\alpha }_{C_1}+{\alpha }_{\mathrm{e}})$
$2^{nd}$ $K$ -Banhatti	$B_2$	$\sum _{\mathrm{e}={\mathrm{C}}_1\sim C_2}({\alpha }_{C_1}\times {\alpha }_{\mathrm{e}})$
$1^{st}$ $K$ -hyper Banhatti	$H{B}_1$	$\sum _{\mathrm{e}={\mathrm{C}}_1\sim C_2}({\alpha }_{C_1}+{\alpha }_{\mathrm{e}}{)}^2$
$2^{nd}$ $K$ -hyper Banhatti	$H{B}_2$	$\sum _{e=C_1\sim {\mathrm{C}}_2}({\alpha }_{C_1}\times {\alpha }_{\mathrm{e}}{)}^2$

Shannon introduced the concept of entropy in 1948,^41–43 which measures the molecular disorganization of the system as well.^19,44 The measurement of edge-weighted graph entropy was initially presented in 2009 ¹⁸ for an edge-weighted graph $G=((V,E),\psi (P_{{\dot{s}}_1}{P}_{{\dot{s}}_2}))$ , where $\psi (P_{{\dot{s}}_1}{P}_{{\dot{s}}_2})$ denotes the edge-weight of an edge $(P_{{\dot{s}}_1}{P}_{{\dot{s}}_2})$ . As demonstrated in a series of papers,^45–49 we perform the entropy calculations based on the scalar multiplicative indices instead of standard multiplicative indices to show the discriminating power of different indices considered in this study. The entropy of a G is defined by the formula:

$${\mathrm{ENT}}_{\psi }(G)=-\sum _{{\dot{s}}_1\sim {\dot{s}}_2}{\displaystyle \frac{\psi (P_{{\dot{s}}_1}{P}_{{\dot{s}}_2})}{{\sum }_{{\dot{s}}_1\sim {\dot{s}}_2}\psi (P_{{\dot{s}}_1}{P}_{{\dot{s}}_2})}\log \left\{{\displaystyle \frac{\psi (P_{{\dot{s}}_1}{P}_{{\dot{s}}_2})}{{\sum }_{{\dot{s}}_1\sim {\dot{s}}_2}\psi (P_{{\dot{s}}_1}{P}_{{\dot{s}}_2})}}\right\}}$$

(1)

Results

In this section, we determine exact values of all the degree-based topological indices and entropies tabulated in Tables 1 and 2. For this, first we provide the edge partition of diamond crystal $\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell }$ as shown in Figures 1 and 2 on the base of degree of vertices and edges in Table 3.

Figure 1.

Step-by-step structure of a diamond.

Figure 2.

Tetrahedral structure of a diamond.

Table 2.

Formulae to compute the entropy on the base of indices tabulated in Table 1.

Entropy	Symbol	Formula
1st redefined Zagreb	$EN{T}_{ReZ{G}_1}$	$\log (EN{T}_{ReZ{G}_1})-\frac{1}{EN{T}_{ReZ{G}_1}}\log \left\{\prod _{C_1\sim C_2}{\left[\frac{{\alpha }_{C_1}+{\alpha }_{C_2}}{{\alpha }_{C_1}{\alpha }_{C_2}}\right]}^{\left[\frac{{\alpha }_{C_1}+{\alpha }_{C_2}}{{\alpha }_{C_1}{\alpha }_{C_2}}\right]}\right\}$
2nd redefined Zagreb	$EN{T}_{ReZ{G}_2}$	$\log (EN{T}_{ReZ{G}_2})-\frac{1}{EN{T}_{ReZ{G}_2}}\log \left\{\prod _{C_1\sim C_2}{\left[\frac{{\alpha }_{C_1}{\alpha }_{C_2}}{{\alpha }_{C_1}+{\alpha }_{C_2}}\right]}^{\left[\frac{{\alpha }_{C_1}{\alpha }_{C_2}}{{\alpha }_{C_1}+{\alpha }_{C_2}}\right]}\right\}$
3rd redefined Zagreb	$EN{T}_{ReZ{G}_3}$	$\log (EN{T}_{ReZ{G}_2})-\frac{1}{EN{T}_{ReZ{G}_2}}\log \left\{\prod _{C_1\sim C_2}{[({\alpha }_{C_1}{\alpha }_{C_2})({\alpha }_{C_1}+{\alpha }_{C_2})]}^{[({\alpha }_{C_1}{\alpha }_{C_2})({\alpha }_{C_1}+{\alpha }_{C_2})]}\right\}$
ABS Connectivity	$EN{T}_{ABS}$	$\log (EN{T}_{ABS})-\frac{1}{EN{T}_{ABS}}\log \left\{\prod _{C_1\sim C_2}{\left[\sqrt{\frac{{\alpha }_{C_1}+{\alpha }_{C_2}-2}{{\alpha }_{C_1}+{\alpha }_{C_2}}}\right]}^{\left[\sqrt{\frac{{\alpha }_{C_1}+{\alpha }_{C_2}-2}{{\alpha }_{C_1}+{\alpha }_{C_2}}}\right]}\right\}$
1st K-Banhatti	$EN{T}_{B_1}$	$\log (EN{T}_{B_1})-\frac{1}{EN{T}_{B_1}}\log \left\{\prod _{e=C_1\sim C_2}{[{\alpha }_{C_1}+{\alpha }_e]}^{[{\alpha }_{C_1}+{\alpha }_e]}\right\}$
2nd K-Banhatti	$EN{T}_{B_2}$	$\log (EN{T}_{B_2})-\frac{1}{EN{T}_{B_2}}\log \left\{\prod _{e=C_1\sim C_2}{[{\alpha }_{C_1}{\alpha }_e]}^{[{\alpha }_{C_1}{\alpha }_e]}\right\}$
1st K-hyper Banhatti	$EN{T}_{H{B}_1}$	$\log (EN{T}_{B_1})-\frac{1}{EN{T}_{B_1}}\log \left\{\prod _{e=C_1\sim C_2}{[{\alpha }_{C_1}+{\alpha }_e]}^{2{[{\alpha }_{C_1}+{\alpha }_e]}^2}\right\}$
2nd K-hyper Banhatti	$EN{T}_{H{B}_2}$	$\log (EN{T}_{B_2})-\frac{1}{EN{T}_{B_2}}\log \left\{\prod _{e=C_1\sim C_2}{[{\alpha }_{C_1}{\alpha }_e]}^{2{[{\alpha }_{C_1}{\alpha }_e]}^2}\right\}$

Table 3.

Edge partition with edge degree of $\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell }$ .

Edge part	$E^{(1,4)}$	$E^{(2,4)}$	$E^{(3,4)}$	$E^{(4,4)}$
Edge degree	3	4	5	6
Frequency	$4$	$12(l-1)$	$6l(l-3)+12$	$\frac{1}{3}(2l^3-12l^2+22l-12)$

By using the edge partition of Table 3 in each formula of Table 1, we have the following equations:

$$ReZ{G}_1(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=(4){\phi }^{\frac{5}{4}}+(12(l-1)){\phi }^{\frac{6}{8}}+(6l(l-3)+12){\phi }^{\frac{7}{12}}+(\lambda ){\phi }^{\frac{8}{16}}$$

(2)

$$ReZ{G}_2(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=(4){\phi }^{\frac{4}{5}}+(12(l-1)){\phi }^{\frac{8}{6}}+(6l(l-3)+12){\phi }^{\frac{12}{7}}+(\lambda ){\phi }^{\frac{16}{8}}$$

(3)

$$ReZ{G}_3(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=(4){\phi }^{20}+(12(l-1)){\phi }^{48}+(6l(l-3)+12){\phi }^{84}+(\lambda ){\phi }^{128}$$

(4)

$$ABS(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=(4){\phi }^{\sqrt{\frac{3}{5}}}+(12(l-1)){\phi }^{\sqrt{\frac{4}{6}}}+(6l(l-3)+12){\phi }^{\sqrt{\frac{5}{7}}}+(\lambda ){\phi }^{\sqrt{\frac{6}{8}}}$$

(5)

$$B_1(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=(4){\phi }^{11}+(12(l-1)){\phi }^{14}+(6l(l-3)+12){\phi }^{17}+(\lambda ){\phi }^{20}$$

(6)

$$B_2(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=(4){\phi }^{15}+(12(l-1)){\phi }^{24}+(6l(l-3)+12){\phi }^{35}+(\lambda ){\phi }^{40}$$

(7)

$$H{B}_1(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=(4){\phi }^{65}+(12(l-1)){\phi }^{100}+(6l(l-3)+12){\phi }^{145}+(\lambda ){\phi }^{200}$$

(8)

$$H{B}_2(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=(4){\phi }^{153}+(12(l-1)){\phi }^{320}+(6l(l-3)+12){\phi }^{625}+(\lambda ){\phi }^{1152}$$

(9)

After evaluating the first derivative of the right hand side of each equation form (2) to (9) at $\phi =1$ , we get the following indices:

$$ReZ{G}_1(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })={\displaystyle \frac{1}{3}{l}^3+{\displaystyle \frac{3}{2}{l}^2+{\displaystyle \frac{13}{6}l+1}}}$$

(10)

$$ReZ{G}_2(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })={\displaystyle \frac{4}{3}{l}^3+{\displaystyle \frac{16}{7}{l}^2-{\displaystyle \frac{4}{21}l-{\displaystyle \frac{8}{35}}}}}$$

(11)

$$ReZ{G}_3(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })={\displaystyle \frac{256}{3}{l}^3-8l^2+{\displaystyle \frac{8}{3}l}}$$

(12)

$$ABS(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=(4)\sqrt{{\displaystyle \frac{3}{5}}}+(12(l-1))\sqrt{{\displaystyle \frac{4}{6}}}+(6l(l-3)+12)\sqrt{{\displaystyle \frac{5}{7}}}+\left({\displaystyle \frac{2l^3-12l^2+22l-12}{3}}\right)\sqrt{{\displaystyle \frac{6}{8}}}$$

(13)

$$B_1(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })={\displaystyle \frac{40}{3}{l}^3+22l^2+{\displaystyle \frac{26}{3}l}}$$

(14)

$$B_2(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=32l^3+18l^2-10l$$

(15)

$$H{B}_1(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })={\displaystyle \frac{400}{3}{l}^3+70l^2+{\displaystyle \frac{170}{3}l}}$$

(16)

$$H{B}_2(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=768l^3-858l^2+1038l-336$$

(17)

Now, by using indices from equations (10) to (17) in formulae of entropy, given in Table 2, we obtain the following entropies:

$$\begin{array}{rl}{\mathrm{ENT}}_{ReZ{G}_1}(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })& =\log \left({\displaystyle \frac{1}{3}{l}^3+{\displaystyle \frac{3}{2}{l}^2+{\displaystyle \frac{13}{6}l+1}}}\right)-{\displaystyle \frac{1}{\left({\displaystyle \frac{1}{3}{l}^3+{\displaystyle \frac{3}{2}{l}^2+{\displaystyle \frac{13}{6}l+1}}}\right)}}\\ & \log \left\{(4){({\displaystyle \frac{5}{4})}}^{\frac{5}{4}}\times (12(l-1)){({\displaystyle \frac{6}{8})}}^{\frac{6}{8}}\times (6l(l-3)+12){({\displaystyle \frac{7}{12})}}^{\frac{7}{12}}\times (\lambda ){({\displaystyle \frac{8}{16})}}^{\frac{8}{16}}\right\}\end{array}$$

$$\begin{array}{rl}{\mathrm{ENT}}_{ReZ{G}_2}(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })& =\log \left({\displaystyle \frac{4}{3}{l}^3+{\displaystyle \frac{16}{7}{l}^2-{\displaystyle \frac{4}{21}l-{\displaystyle \frac{8}{35}}}}}\right)-{\displaystyle \frac{1}{{\displaystyle \frac{4}{3}{l}^3+{\displaystyle \frac{16}{7}{l}^2-{\displaystyle \frac{4}{21}l-{\displaystyle \frac{8}{35}}}}}}}\\ & \log \left\{(4){({\displaystyle \frac{4}{5})}}^{\frac{4}{5}}\times (12(l-1)){({\displaystyle \frac{8}{6})}}^{\frac{8}{6}}\times (6l(l-3)+12){({\displaystyle \frac{12}{7})}}^{\frac{12}{7}}\times (\lambda ){({\displaystyle \frac{16}{8})}}^{\frac{16}{8}}\right\}\end{array}$$

$$\begin{array}{rl}{\mathrm{ENT}}_{ReZ{G}_3}(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })& =\log \left({\displaystyle \frac{256}{3}{l}^3-8l^2+{\displaystyle \frac{8}{3}l}}\right)-{\displaystyle \frac{1}{{\displaystyle \frac{256}{3}{l}^3-8l^2+{\displaystyle \frac{8}{3}l}}}\log \{(4){(20)}^{20}}\\ & \times (12(l-1)){48}^{48}\times (6l(l-3)+12){84}^{84}\times (\lambda ){128}^{128}\}\end{array}$$

$$\begin{array}{rl}{\mathrm{ENT}}_{\mathrm{ABS}}(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })& =\log (\mathrm{ABS})-{\displaystyle \frac{1}{\mathrm{ABS}}\log \{(4){\left(\sqrt{{\displaystyle \frac{3}{5}}}\right)}^{\sqrt{\frac{3}{5}}}\times (12(l-1)){\left(\sqrt{{\displaystyle \frac{4}{6}}}\right)}^{\sqrt{\frac{4}{6}}}}\\ & \times (6l(l-3)+12){\left(\sqrt{{\displaystyle \frac{5}{7}}}\right)}^{\sqrt{\frac{5}{7}}}\times (\lambda ){\left(\sqrt{{\displaystyle \frac{6}{8}}}\right)}^{\sqrt{\frac{6}{8}}}\}\end{array}$$

$$\begin{array}{rl}{\mathrm{ENT}}_{B_1}(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell }& =\log \left({\displaystyle \frac{40}{3}{l}^3+22l^2+{\displaystyle \frac{26}{3}l}}\right)-{\displaystyle \frac{1}{{\displaystyle \frac{40}{3}{l}^3+22l^2+{\displaystyle \frac{26}{3}l}}}}\\ & \log \{(4){(11)}^{11}\times (12(l-1)){(14)}^{14}\times (6l(l-3)+12){(17)}^{17}\times (\lambda ){(20)}^{20}\}\end{array}$$

$$\begin{array}{rl}{\mathrm{ENT}}_{B_2}(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell }& =\log (32l^3+18l^2-10l)-{\displaystyle \frac{1}{32l^3+18l^2-10l}\log \{(4){15}^{15}}\\ & \times (12(l-1)){24}^{24}\times (6l(l-3)+12){35}^{35}\times (\lambda ){40}^{40}\}\end{array}$$

$$\begin{array}{rl}{\mathrm{ENT}}_{{\mathrm{HB}}_1}(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })& =\log \left({\displaystyle \frac{400}{3}{l}^3+70l^2+{\displaystyle \frac{170}{3}l}}\right)-{\displaystyle \frac{1}{{\displaystyle \frac{400}{3}{l}^3+70l^2+{\displaystyle \frac{170}{3}l}}}\log \{(4)({65}^{65})}\\ & (12(l-1))({100}^{100})\times (6l(l-3)+12)({145}^{145})\times (\lambda )({200}^{200})\}\end{array}$$

$$\begin{array}{rl}& {\mathrm{ENT}}_{{\mathrm{HB}}_2}(\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell })=\log (768l^3-858l^2+1038l-336)-{\displaystyle \frac{1}{768l^3-858l^2+1038l-336}}\\ & \log \{(4){(153)}^{153}\times (12(l-1)){320}^{320}\times (6l(l-3)+12){625}^{625}\times (\lambda ){1152}^{1152}\}\end{array}$$

Comparison

This part presents numeric and graphical comparisons of the computed results as presented in Table 4 and their graphical analysis as shown in Figure 3. For the sake of comparison of results, we take values $l=2,3,\dots ,12$ . We observed that as the value of l increased the index and entropy values are also increased gradually. We utilize Corel Draw and Origin for figure creation, and Maple for numeric computations.

Figure 3.

Graphical expression of TIs of $\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell }$ .

Table 4.

Numeric expressions of TIs of $\mathcal{C}\mathcal{B}{\mathcal{R}}_{\ell }$ .

$l$	ReZG1	ReZG2	ReZG3	ABS	B1	B2	HB1	HB2
2	14	19.2	656	12.896	212	348	1460	4452
3	30	55.77	2240	32.84	584	1056	4400	15792
4	55	120.91	5344	66.382	1240	2376	9880	39240
5	91	222.63	10480	116.998	2260	4500	18700	79404
6	140	368.91	18160	188.15	3724	7620	31660	140892
7	204	567.77	28896	283.296	5712	11928	49560	228312
8	285	827.2	43200	405.91	8304	17616	73200	346272
9	385	1155.2	61584	559.44	11580	24876	103380	499380
10	506	1559.77	84560	747.37	15620	33900	140900	692244
11	650	2048.91	112640	973.15	20504	44880	186560	929472
12	819	2630.63	146336	1240.25	26312	58008	241160	1215672

Concluding remarks

This manuscript presents key findings regarding the entropy of eight degree-based topological indices for a molecular structure of diamonds. By the reason of rich conception, we presented numeric and graphical analysis of obtained results of toplogical indices for few initial values of the used parameter as shown in Table 4 and Figure 3. From obtained results and their provided comparisons, each of the calculated topological indices can be regarded as a finest/moral or nastiest/immoral descriptor for a topological characterization for the molecular structure of diamond according to the following values-based hierarchy (this pyramid is just based upon our mathematical computations, numeric and graphical analysis. However, the novelty of each topological index can be followed after investigating that which index is best correlating with which of the chemical property):

$$ReZ{G}_1<ABS<ReZ{G}_2<B_1<B_2<ReZ{G}_3<H{B}_1<H{B}_2.$$