Topological Study on Triazine-Based Covalent-Organic Frameworks

Abstract

Most of the research has evidenced that there is a strong natural correlation among the chemical properties of molecular structures. This study analyses supramolecular chemistry and investigates topological indices of supramolecular structures called triazine-based covalent-organic frameworks. The use of degree-based topological indices on these chemical molecular structures can aid material scientists in better understanding their chemical and biological properties, thus compensating for the lack of chemical tests. This study aims to theoretically examine the triazine-based covalent-organic frameworks (TriCF) utilizing degree-based topological indices, specifically multiplicative topological indices and entropy measures. A detailed comparison of the computed topological indices of the aforementioned chemical structures is described using graphical depiction.

1. Introduction

Chemical graph theory integrates graph theory and chemistry in an area of mathematics. After obtaining insight into the physical features of chemical substances, graph theory is employed mathematically to model them [1]. For centuries, graph theory and chemistry have been interlinked and well connected. Various investigations in both disciplines have created extraordinarily strong linkages between mathematics and chemistry, resulting in the emergence of a scientific field known as chemical graph theory. The first instances of chemical graphs are from the late eighteenth century, when Isaac Newton’s ideas influenced chemistry’s perspectives. Even though research into atom interactions accelerated during that century, chemical bonds were not discovered [1]. As a result, the initial application of chemical graphs was to depict hypothetical forces between molecules and atoms [2].

In this study, we consider a triazine-based covalent organic polymer (TriCF) as a chemical graph. The TriCF structure is one of the supermolecular families in the field of supermolecular chemistry [3]. Supramolecular chemistry, or the “chemistry of molecular assemblages and the intermolecular connection”, is one of the most popular and rapidly growing disciplines of chemistry, emphasising going “above” molecular chemistry. It is the study of systems that comprise several molecular assemblies with the goal of understanding their structure, function and attributes. Supramolecular chemistry investigates molecular self-assembly, protein folding, molecular recognition, host-guest chemistry, mechanically interlocked molecular architectures and dynamic covalent chemistry, among other phenomena. It is highly interdisciplinary in nature, attracting biologists, environmental scientists, physicists, biochemists, theoreticians and crystallographers in addition to chemists. Supramolecular chemistry’s potential to lead to modern technologies has sparked recent research [3]. Supramolecular chemistry is still a young field that is rapidly evolving thanks to contributions from the range of fields given above. Supramolecular chemistry emerged at the turn of the century as a breakthrough approach to deal with chemical substances and concepts, utilising capabilities beyond the bounds of single molecules. Supramolecular chemistry has already demonstrated the enormous potential for designing intelligent molecules and gadgets. The next generation of advanced machines is projected to be based on molecular electronics and photonics. This field has recently focused on the development of molecular devices capable of sensing, photo switching, separation, motion, and transport [3,4,5].

The triazines have a six-membered planar benzene-like ring with three nitrogens replacing the carbons. The locations of the nitrogen atoms distinguish the three isomers of triazine [6]. Two well-known triazine structures are melamine and cyanuric chloride. Melamine, a white, odourless, crystalline, and nontoxic hetero-aromatic molecule, is a chemical commodity used as a raw ingredient in a variety of everyday products that must meet stringent durability and wear resistance standards [2,6]. Cyanuric chloride (2,4,6-trichloro-1,3,5-triazine) is a versatile multifunctional reagent that has been widely employed in bioconjugation as well as the production of insecticides, brighteners and reactive dyes. Triazine-based covalent organic polymers (COPs) are a growing sub-class of porous organic framework materials made from triazine or nitrile-containing precursors by covalent bonding. The synthesis of triazine-based covalent-organic frameworks (see Figure 1) using melamine and cyanuric chloride is discussed by Wen et al. [6]. TriCF denotes the as-prepared covalent-organic frameworks. TriCF has a lamellar structure and is extremely thermo-stable. It is a novel synthesized brand-new lubricant [6]. There are currently no studies in the literature on the computation of topological indices of TriCF’s structure. The results of this study could be applied to future triazine-based covalent organic frameworks that are designed and manufactured [6].

The topological index of a molecule structure is a numerical quantity that quantifies the molecular structure and its branching pattern in a non-empirical way [7]. In this juncture, the topological index can be thought of as a score function that converts each chemical structure into a real number and then uses that number as a descriptor for the molecule under investigation. For grasping the links between the molecular structure and the prospective physicochemical features, chemical engineers use various well-known indices. The topological indices may be used in drug design, molecular QSPR/QSAR analysis, thermodynamics and other fields [4,5,6,7,8]. For instance, it has been demonstrated that there is a strong association between the boiling points and heats of production of specific classes of isomeric octanes and the atom bond connectivity index. Chemical compounds, materials and pharmaceuticals are represented as (molecular) graphs in theoretical chemistry, with each vertex representing an atom of the molecule structure and each edge implying covalent bonds between two atoms [8].

In an effort to describe the complexity of graphs, graph entropy was developed. Although it was initially developed to illustrate the complexity of information transmission and communication, it currently has numerous important applications in a variety of scientific fields, including physical dissipative structures, biological systems, engineering fields and others [8]. Graph entropies can be divided into two categories: deterministic and probabilistic [7,8,9,10]. Since it is frequently used in a variety of fields, including communication and the characterisation of chemical structures, this work focuses on the probabilistic category [9,11]. Statistical techniques are further divided into intrinsic and extrinsic groups. In intrinsic measures, a graph is divided into parts with similar structural similarities, and a probability distribution over those parts is discovered. For extrinsic measurements, vertices or edges in the graph are given a probability function [11]. Applying an entropy function to this probability distribution function yields the numerical value of probabilistic measurements of graph complexity [7,8,9,10,11,12,13].

2. Preliminaries and Mathematical Terminologies

In this paper, we consider $\gamma$ to be a connected graph. The vertex set and edge set are denoted by the symbols $\left|V\right(\gamma \left)\right|=\mathcal{P}$ and $\left|E\right(\gamma \left)\right|=\mathcal{Q}$, respectively. In order to create multiplicative topological indices and degree-based entropy measurements for the TriCF structure, the research uses edge-partition techniques.

The research shows various degree-based topological descriptors and multiplicative degree-based topological descriptors, which can be seen in

Table 1. Degree-based topological indices.

Randić Index	${\displaystyle R\left(G\right)=\sum _{uv\in E\left(G\right)}^{}\left[{\displaystyle \frac{1}{\sqrt{d_u{d}_v}}}\right]}$
Reciprocal Randić Index	${\displaystyle RR\left(G\right)=\sum _{uv\in E\left(G\right)}^{}\left[\sqrt{d_u{d}_v}\right]}$
Reduced Reciprocal Randić Index	${\displaystyle RRR\left(G\right)=\sum _{uv\in E\left(G\right)}^{}\left[\sqrt{(d_u-1)(d_v-1)}\right]}$
First Zagreb Index	$M_1\left(G\right)$=${\displaystyle \sum _{uv\in E\left(G\right)}}$$[d_u+d_v]$
Second Zagreb Index	${\displaystyle M_2\left(G\right)=\sum _{uv\in E\left(G\right)}[d_u\times d_v]}$
Reduced Second Zagreb Index	${\displaystyle R{M}_2\left(G\right)=\sum _{uv\in E\left(G\right)}^{}\left[(d_u-1)(d_v-1)\right]}$
Hyper Zagreb Index	${\displaystyle HM\left(G\right)=\sum _{uv\in E\left(G\right)}^{}{[d_u+d_v]}^2}$
Augmented Zagreb Index	${\displaystyle AZ\left(G\right)=\sum _{uv\in E\left(G\right)}^{}{\left[{\displaystyle \frac{d_u\times d_v}{d_u+d_v-2}}\right]}^3}$
Harmonic Index	${\displaystyle H\left(G\right)=\sum _{uv\in E\left(G\right)}^{}\left[{\displaystyle \frac{2}{d_u+d_v}}\right]}$
Sum Connectivity Index	${\displaystyle SC\left(G\right)=\sum _{uv\in E\left(G\right)}^{}{\displaystyle \frac{1}{\sqrt{d_u+d_v}}}}$
Geometric Arithmetic Index	${\displaystyle GA\left(G\right)=\sum _{uv\in E\left(G\right)}^{}{\displaystyle \frac{2\sqrt{d_u{d}_v}}{d_u+d_v}}}$
Inverse Sum Index	${\displaystyle IS\left(G\right)=\sum _{uv\in E\left(G\right)}^{}{\displaystyle \frac{d_u{d}_v}{d_u+d_v}}}$
Albertson Index	$Alb\left(G\right)$=${\displaystyle \sum _{uv\in E\left(G\right)}}$$|u-v|$
Symmetric Division Index	${\displaystyle AZI\left(G\right)=\sum _{uv\in E\left(G\right)}^{}{\displaystyle \frac{{d_u}^2+{d_v}^2}{d_u{d}_v}}}$
Atom Bond Connectivity Index	${\displaystyle ABC\left(G\right)=\sum _{uv\in E\left(G\right)}^{}\sqrt{{\displaystyle \frac{d_u+d_v-2}{d_u{d}_v}}}}$

[10,11,12,13,14,15,16,17,18,19,20,21,22] and

Table 2. Multiplicative degree-based topological indices.

Randić Index	${\displaystyle R\left(G\right)=\prod _{uv\in E\left(G\right)}^{}\left[{\displaystyle \frac{1}{\sqrt{d_u{d}_v}}}\right]}$
Reciprocal Randić Index	${\displaystyle RR\left(G\right)=\prod _{uv\in E\left(G\right)}^{}\left[\sqrt{d_u{d}_v}\right]}$
Reduced Reciprocal Randić Index	${\displaystyle RRR\left(G\right)=\prod _{uv\in E\left(G\right)}^{}\left[\sqrt{(d_u-1)(d_v-1)}\right]}$
First Zagreb Index	$M_1\left(G\right)$=${\displaystyle \prod _{uv\in E\left(G\right)}}$$[d_u+d_v]$
Second Zagreb Index	${\displaystyle M_2\left(G\right)=\prod _{uv\in E\left(G\right)}[d_u\times d_v]}$
Reduced Second Zagreb Index	${\displaystyle R{M}_2\left(G\right)=\prod _{uv\in E\left(G\right)}^{}\left[(d_u-1)(d_v-1)\right]}$
Hyper Zagreb Index	${\displaystyle HM\left(G\right)=\prod _{uv\in E\left(G\right)}^{}{[d_u+d_v]}^2}$
Augmented Zagreb Index	${\displaystyle AZ\left(G\right)=\prod _{uv\in E\left(G\right)}^{}{\left[{\displaystyle \frac{d_u\times d_v}{d_u+d_v-2}}\right]}^3}$
Harmonic Index	${\displaystyle H\left(G\right)=\prod _{uv\in E\left(G\right)}^{}\left[{\displaystyle \frac{2}{d_u+d_v}}\right]}$
Sum Connectivity Index	${\displaystyle SC\left(G\right)=\prod _{uv\in E\left(G\right)}^{}{\displaystyle \frac{1}{\sqrt{d_u+d_v}}}}$
Geometric Arithmetic Index	${\displaystyle GA\left(G\right)=\prod _{uv\in E\left(G\right)}^{}{\displaystyle \frac{2\sqrt{d_u{d}_v}}{d_u+d_v}}}$
Inverse Sum Index	${\displaystyle IS\left(G\right)=\prod _{uv\in E\left(G\right)}^{}{\displaystyle \frac{d_u{d}_v}{d_u+d_v}}}$
Albertson Index	$Alb\left(G\right)$=${\displaystyle \prod _{uv\in E\left(G\right)}}$$|u-v|$
Symmetric Division Index	${\displaystyle AZI\left(G\right)=\prod _{uv\in E\left(G\right)}^{}{\displaystyle \frac{{d_u}^2+{d_v}^2}{d_u{d}_v}}}$
Atom Bond Connectivity Index	${\displaystyle ABC\left(G\right)=\prod _{uv\in E\left(G\right)}^{}\sqrt{\frac{d_u+d_v-2}{d_u{d}_v}}}$

[23,24,25,26,27,28,29,30,31], respectively.

3. Methodology

The computations in this study are carried out using graph theoretical technologies, namely the edge partition method, and analytical methods. The molecular structures of the TriCF are described using Chem Draw Ultra, and numerical results are visualized using Origin.

4. Main Results

In this section, the authors present key findings of the study. A schematic of triazine-based covalent-organic frameworks synthesis can be formed into any kind of structure, such as a linear chain, parellelogram, hexagonal shape, etc. Linear chain triazine-based covalent-organic frameworks, parellelogram triazine-based covalent-organic frameworks, and hexagonal triazine-based covalent-organic frameworks are here denoted as TriCF(r), TriCF(r,s), and TriCF(r), respectively. Figure 2, Figure 3 and Figure 4 show the 2D structure of linear chain, parellogram, hexagonal TriCF structures respectively. Figure 5 shows the unit cell of TriCF and explains the edge partition method. In this work, based on the growth of TriCF, the authors categorized it into three kinds. In particular, ${\gamma }_1$, ${\gamma }_2$ and ${\gamma }_3$ are linear chain TriCF structures, parellelogram TriCF structures and hexagonal TriCF structures, respectively.

• The total number of vertices and edges of ${\gamma }_1$ is depicted as $\mathcal{P}$ = $33r+15$ and $\mathcal{Q}$ = $36r+18$, respectively. In addition, the edge set has two edge partitions. The first edge partition contains edges with the following values: $2r+4$; ${𝒹ℊ}_1$$\left(u\right)$= 1, ${𝒹ℊ}_1$$\left(v\right)$= 3 and $34r+14$ edges make up the second edge partition, where ${𝒹ℊ}_1$$\left(u\right)$= 2, ${𝒹ℊ}_1$$\left(v\right)$= 3.
• The total number of vertices and edges of ${\gamma }_2$ is depicted as $\mathcal{P}$ = $(13s+20)r+18s+3$ and $\mathcal{Q}$ = $(18s+18)r+18s$, respectively. In addition, the edge set has two edge partitions. The first edge partition contains edges with the following values: $2r+2s+2$; ${𝒹ℊ}_{\mathcal{2}}$$\left(u\right)$= 1, ${𝒹ℊ}_{\mathcal{2}}$$\left(v\right)$= 3 and $(18s+16)r+16s-2$ edges make up the second edge partition, where ${𝒹ℊ}_{\mathcal{2}}$$\left(u\right)$= 2,${𝒹ℊ}_{\mathcal{2}}$$\left(v\right)$= 3.
• The total number of vertices and edges of ${\gamma }_3$ is depicted as $\mathcal{P}$ = $45r^2+3r$ and $\mathcal{Q}$ = $54r^2$, respectively. In addition, the edge set has two edge partitions. The first edge partition contains edges with the following values: $6r$; ${𝒹ℊ}_{\mathcal{3}}$$\left(u\right)$= 1, ${𝒹ℊ}_{\mathcal{3}}$$\left(v\right)$= 3 and $54r^2-6r$ edges make up the second edge partition, where ${𝒹ℊ}_{\mathcal{3}}$$\left(u\right)$= 2, ${𝒹ℊ}_{\mathcal{3}}$$\left(v\right)$= 3.

Table 3. Edge partition of $\gamma$.

$\mathit{\gamma }$	$(1,3)$	$(2,3)$	$\mathcal{Q}( )$	$\mathcal{P}( )$
${\gamma }_1$	$2r+4$	$34r+14$	$36r+18$	$36r+15$
${\gamma }_2$	$2r+(2s+2)$	$(18s+16)r+16s-2$	$(18s+18)r+18s$	$(13s+20)r+18s+3$
${\gamma }_3$	$6r$	$54r^2-6r$	$54r^2$	$45r^2+3r$

below illustrates how the structure’s edges are divided.

4.1. Degree-Based Topological Indices of TriCF Structure

Theorem 1.

Let ${\gamma }_1$ be a linear chain TriCF structure with dimension r. Then,

1. 

$R\left({\gamma }_1\right)={\displaystyle \frac{\sqrt{3}\left(17\sqrt{2}r+7\sqrt{2}+2r+4\right)}{3}}$.

2. 

$RR\left({\gamma }_1\right)=2\sqrt{3}\left(17\sqrt{2}r+7\sqrt{2}+r+2\right)$.

3. 

$RRR\left({\gamma }_1\right)=\sqrt{2}\left(34r+14\right)$.

4. 

$M_1\left({\gamma }_1\right)=178r+86$.

5. 

$M_2\left({\gamma }_1\right)=210r+96$.

6. 

$R{M}_2\left({\gamma }_1\right)=68r+28$.

7. 

$HM\left({\gamma }_1\right)=882r+414$.

8. 

$AZ\left({\gamma }_1\right)={\displaystyle \frac{1115r}{4}}+{\displaystyle \frac{251}{2}}$.

9. 

$H\left({\gamma }_1\right)={\displaystyle \frac{73r}{5}}+{\displaystyle \frac{38}{5}}$.

10. 

$SC\left({\gamma }_1\right)=r+2+{\displaystyle \frac{\sqrt{5}\left(34r+14\right)}{5}}$.

11. 

$GA\left({\gamma }_1\right)={\displaystyle \frac{1}{5}}\left(2\sqrt{3}\left(34\sqrt{10}r+14\sqrt{10}+5r+10\right)\right)$.

12. 

$IS\left({\gamma }_1\right)={\displaystyle \frac{423r}{10}}+{\displaystyle \frac{99}{5}}$.

13. 

$AZI\left({\gamma }_1\right)={\displaystyle \frac{241r}{3}}+{\displaystyle \frac{131}{3}}$.

14. 

$Alb\left({\gamma }_1\right)=38r+22$.

15. 

$ABC\left({\gamma }_1\right)={\displaystyle \frac{2\sqrt{3r+6}}{3}}+\sqrt{17r+7}$.

Proof. 

Let ${\gamma }_1$ be a linear chain TriCF structure with $\mathcal{P}( )$ and $\mathcal{Q}( )$ (see Table 3). We have conducted edge partitions of ${\gamma }_1$ based on the vertex degree, and the following results are obtained by applying those edge partitions in the definitions of degre- based topological indices (Table 1).

1. 

$R\left({\gamma }_1\right)={\displaystyle \frac{1}{\sqrt{1\times 3}}}(2r+4)+{\displaystyle \frac{1}{\sqrt{2\times 3}}}(34r+14)$

$={\displaystyle \frac{\sqrt{3}(2r+4)}{3}}+{\displaystyle \frac{\sqrt{6}(34r+14)}{6}}$

$={\displaystyle \frac{\sqrt{3}(17\sqrt{2}r+7\sqrt{2}+2r+4)}{3}}$.

2. 

$RR\left({\gamma }_1\right)=\sqrt{1\times 3}(2r+4)+\sqrt{2\times 3}(34r+14)$$=2\sqrt{3}\left(17\sqrt{2}r+7\sqrt{2}+r+2\right)$.

3. 

$RRR\left({\gamma }_1\right)=\sqrt{0}(2r+4)+\sqrt{2}(34r+14)$

$=\sqrt{2}(34r+14)$.

4. 

$M_1\left({\gamma }_1\right)=4(2r+4)+5(34r+14)$

$=178r+86$.

5. 

$M_2\left({\gamma }_1\right)=3(2r+4)+\left(6\right)(34r+14)$

$=210r+96$.

6. 

$R{M}_2\left({\gamma }_1\right)=\left(0\right)\left(2\right)(2r+4)+\left(1\right)\left(2\right)(34r+14)$

$=68r+28$.

7. 

$HM\left({\gamma }_1\right)={(1+3)}^2(2r+4)+{(2+3)}^2(34r+14)$

$=882r+414$.

8. 

$AZ\left({\gamma }_1\right)={\left({\displaystyle \frac{1\times 3}{(1+3-2)}}\right)}^3(2r+4)+{\left({\displaystyle \frac{2\times 3}{(2+3-2)}}\right)}^3(34r+14)$

$={\displaystyle \frac{1115r}{4}}+{\displaystyle \frac{251}{2}}$.

9. 

$H\left({\gamma }_1\right)={\displaystyle \frac{2}{4}}(2r+4)+{\displaystyle \frac{2}{5}}(34r+14)$

$={\displaystyle \frac{73r}{5}}+{\displaystyle \frac{38}{5}}$.

10. 

$SC\left({\gamma }_1\right)={\displaystyle \frac{1}{\sqrt{1+3}}}(2r+4)+{\displaystyle \frac{1}{\sqrt{2+3}}}(34r+14)$

$=r+2+{\displaystyle \frac{\sqrt{5}(34r+14)}{5}}$.

11. 

$GA\left({\gamma }_1\right)=2\left({\displaystyle \frac{\sqrt{1\times 3}}{1+3}}\right)(2r+4)+2\left({\displaystyle \frac{\sqrt{2\times 3}}{2+3}}\right)(34r+14)$

$=\sqrt{3}(2r+4)+{\displaystyle \frac{2\sqrt{30}(34r+14)}{5}}$

$={\displaystyle \frac{2\sqrt{3}(34\sqrt{10}r+14\sqrt{10}+5r+10)}{5}}$.

12. 

$IS\left({\gamma }_1\right)={\displaystyle \frac{1\times 3}{1+3}}(2r+4)+{\displaystyle \frac{2\times 3}{2+3}}(34r+14)$

$={\displaystyle \frac{423r}{10}}+{\displaystyle \frac{99}{5}}$.

13. 

$AZI\left({\gamma }_1\right)=\left({\displaystyle \frac{1+9}{1\times 3}}\right)(2r+4)+\left({\displaystyle \frac{4+9}{2\times 3}}\right)(34r+14)$

$={\displaystyle \frac{241r}{3}}+{\displaystyle \frac{131}{3}}$.

14. 

$Alb\left({\gamma }_1\right)=2(2r+4)+1(34r+14)$

$=38r+22$.

15. 

$ABC\left({\gamma }_1\right)=\sqrt{{\displaystyle \frac{1+3-2}{1\times 3}}}(2r+4)+\sqrt{{\displaystyle \frac{2+3-2}{2\times 3}}}(34r+14))$

$={\displaystyle \frac{2\sqrt{3r+6}}{3}}+\sqrt{17r+7}$.

□

Theorem 2.

Let ${\gamma }_2$ be a parallelogram TriCF structure with dimensions r and s. Then, the degree-based topological indices are:

1. 

$R\left({\gamma }_2\right)={\displaystyle \frac{((9s+8)r+8s-1)\sqrt{3}\sqrt{2}}{3}}+{\displaystyle \frac{\sqrt{3}(2r+2s+2)}{3}}$.

2. 

$RR\left({\gamma }_2\right)=((18s+16)r+16s-2)\sqrt{3}\sqrt{2}+2\sqrt{3}(r+s+1)$.

3. 

$RRR\left({\gamma }_2\right)=\sqrt{2}((18s+16)r+16s-2)$.

4. 

$M_1\left({\gamma }_2\right)=(90s+88)r+88s-2$.

5. 

$M_2\left({\gamma }_2\right)=(108s+102)r+102s-6$.

6. 

$R{M}_2\left({\gamma }_2\right)=(36s+32)r+32s-4$.

7. 

$HM\left({\gamma }_2\right)=(450s+432)r+432s-18$.

8. 

$AZ\left({\gamma }_2\right)={\displaystyle \frac{(576r+539)s}{4}}+\frac{539r}{4}-{\displaystyle \frac{37}{4}}$.

9. 

$H\left({\gamma }_2\right)={\displaystyle \frac{(36s+37)r}{5}}+{\displaystyle \frac{37s}{5}}+{\displaystyle \frac{1}{5}}$.

10. 

$SC\left({\gamma }_2\right)=r+s+1+{\displaystyle \frac{\sqrt{5}((18s+16)r+16s-2)}{5}}$.

11. 

$GA\left({\gamma }_2\right)={\displaystyle \frac{((36s+32)r+32s-4)\sqrt{3}\sqrt{10}}{5}}+2\sqrt{3}(r+s+1)$

12. 

$IS\left({\gamma }_2\right)={\displaystyle \frac{(216s+207)r}{10}}+{\displaystyle \frac{207s}{10}}-{\displaystyle \frac{9}{10}}$.

13. 

$Alb\left({\gamma }_2\right)=(18s+20)r+20s+2$.

14. 

$AZI\left({\gamma }_2\right)={\displaystyle \frac{(117r+124)s}{3}}+{\displaystyle \frac{124r}{3}}+\frac{7}{3}$.

15. 

$ABC\left({\gamma }_2\right)={\displaystyle \frac{2\sqrt{3r+3s+3}}{3}}+\sqrt{(9s+8)r+8s-1}$.

Proof. 

Let ${\gamma }_2$ be a parallelogram TriCF structure with $\mathcal{P}( )$ and $\mathcal{Q}( )$ as taken from Table 3. We have conducted edge partitions of ${\gamma }_2$ based on the vertex degree, and the following results are obtained by applying those edge partitions in the definitions of degree-based topological indices (Table 1).

1. 

$\left({\gamma }_2\right)={\displaystyle \frac{1}{\sqrt{1\times 3}}}(2r+2s+2)+{\displaystyle \frac{1}{\sqrt{2\times 3}}}((18s+16)r+16s-2)$

$=\frac{\sqrt{3}(2r+2s+2)}{3}+{\displaystyle \frac{\sqrt{6}((18s+16)r+16s-2)}{6}}$

$={\displaystyle \frac{((9s+8)r+8s-1)\sqrt{3}\sqrt{2}}{3}}+{\displaystyle \frac{\sqrt{3}(2r+2s+2)}{3}}$.

2. 

$RR\left({\gamma }_2\right)=\sqrt{1\times 3}(2r+2s+2)+\sqrt{2\times 3}((18s+16)r+16s-2)$

$=\sqrt{3}(2r+2s+2)+\sqrt{6}((18s+16)r+16s-2)$

$=((18s+16)r+16s-2)\sqrt{3}\sqrt{2}+2\sqrt{3}(r+s+1)$.

3. 

$RRR\left({\gamma }_2\right)=\sqrt{(1-1)(3-1)}(2r+2s+2)+\sqrt{(2-1)(3-1)}((18s+16)r+16s-2)$

$=\sqrt{2}((18s+16)r+16s-2)$.

4. 

$M_1\left({\gamma }_2\right)=(1+3)(2r+2s+2)+(2+3)((18s+16)r+16s-2)$

$=8r+88s-2+5(1s+16)r$

$=(90s+88)r+88s-2$.

5. 

$M_2\left({\gamma }_2\right)=(1\times 3)(2r+2s+2)+(2\times 3)((18s+16)r+16s-2)$

$=6r+102s-6+6(18s+16)r$

$=(108s+102)r+102s-6$.

6. 

$R{M}_2\left({\gamma }_2\right)=\left((1-1)(3-1)\right)(2r+2s+2)+\left((2-1)(3-1)\right)((18s+16)r+16s-2)$

$=2(18s+16)r+32s-4$

$=(36s+32)r+32s-4$.

7. 

$HM\left({\gamma }_2\right)=\left({(1+3)}^2\right)(2r+2s+2)+\left({(2+3)}^2\right)((18s+16)r+16s-2)$

$=32r+432s-18+25(18s+16)r$

$=(450s+432)r+432s-18$.

8. 

$AZ\left({\gamma }_2\right)={\left({\displaystyle \frac{1\times 3}{(1+3-2)}}\right)}^3(2r+2s+2)+{\left({\displaystyle \frac{2\times 3}{(2+3-2)}}\right)}^3((18s+16)r+16s-2)$

$={\displaystyle \frac{27r}{4}}+{\displaystyle \frac{539s}{4}}-{\displaystyle \frac{37}{4}}+8(18s+16)r$

$={\displaystyle \frac{(576r+539)s}{4}}+{\displaystyle \frac{539r}{4}}-{\displaystyle \frac{37}{4}}$.

9. 

$H\left({\gamma }_2\right)=\left({\displaystyle \frac{2}{1+3}}\right)(2r+2s+2)+\left({\displaystyle \frac{2}{2+3}}\right)((18s+16)r+16s-2)$

$=r+{\displaystyle \frac{37s}{5}}+{\displaystyle \frac{1}{5}}+{\displaystyle \frac{2(18s+16)r}{5}}$

$={\displaystyle \frac{(36s+37)r}{5}}+{\displaystyle \frac{37s}{5}}+{\displaystyle \frac{1}{5}}$.

10. 

$SC\left({\gamma }_2\right)={\displaystyle \frac{1}{\sqrt{1+3}}}(2r+2s+2)+{\displaystyle \frac{1}{\sqrt{2+3}}}((18s+16)r+16s-2)$

$=r+s+1+{\displaystyle \frac{\sqrt{5}((18s+16)r+16s-2)}{5}}$.

11. 

$GA\left({\gamma }_2\right)=2\left({\displaystyle \frac{\sqrt{1\times 3}}{1+3}}\right)(2r+2s+2)+2\left({\displaystyle \frac{\sqrt{2\times 3}}{2+3}}\right)((18s+16)r+16s-2)$

$=\sqrt{3}(2r+2s+2)+\frac{2\sqrt{30}((18s+16)r+16s-2)}{5}$

$={\displaystyle \frac{((36s+32)r+32s-4)\sqrt{3}\sqrt{10}}{5}}+2\sqrt{3}(r+s+1)$.

12. 

$IS\left({\gamma }_2\right)=\left({\displaystyle \frac{1\times 3}{1+3}}\right)(2r+2s+2)+\left({\displaystyle \frac{2\times 3}{2+3}}\right)((18s+16)r+16s-2)$

$={\displaystyle \frac{3r}{2}}+{\displaystyle \frac{207s}{10}}-{\displaystyle \frac{9}{10}}+{\displaystyle \frac{6(18s+16)r}{5}}$

$={\displaystyle \frac{(216s+207)r}{10}}+{\displaystyle \frac{207s}{10}}-{\displaystyle \frac{9}{10}}$.

13. 

$Alb\left({\gamma }_2\right)=2(2r+2s+2)+1((18s+16)r+16s-2)$

$=4r+2s+2+(18s+16)r$

$=(18s+20)r+20s+2$.

14. 

$AZI\left({\gamma }_2\right)=\left({\displaystyle \frac{1+9}{1\times 3}}\right)(2r+2s+2)+\left({\displaystyle \frac{4+9}{2\times 3}}\right)((18s+16)r+16s-2)$

$={\displaystyle \frac{20r}{3}}+{\displaystyle \frac{124s}{3}}+{\displaystyle \frac{7}{3}}+{\displaystyle \frac{13(18s+16)r}{6}}$

$={\displaystyle \frac{(117r+124)s}{3}}+{\displaystyle \frac{124r}{3}}+{\displaystyle \frac{7}{3}}$.

15. 

$ABC\left({\gamma }_2\right)=\sqrt{{\displaystyle \frac{1+3-2}{1\times 3}}}(2r+2s+2)+\sqrt{{\displaystyle \frac{2+3-2}{2\times 3}}}((18s+16)r+16s-2)$

$={\displaystyle \frac{2\sqrt{3r+3s+3}}{3}}+\sqrt{{\displaystyle \frac{(18s+16)r}{2}}+8s-1}$

$={\displaystyle \frac{2\sqrt{3r+3s+3}}{3}}+\sqrt{(9s+8)r+8s-1}$.

□

Theorem 3.

Let ${\gamma }_3$ be a hexagonal TriCF structure with dimension r. Then,

1. 

$R\left({\gamma }_3\right)=\sqrt{3}r(9\sqrt{2}r-\sqrt{2}+2)$.

2. 

$RR\left({\gamma }_3\right)=6\sqrt{3}r(9\sqrt{2}r-\sqrt{2}+1)$.

3. 

$RRR\left({\gamma }_3\right)=\sqrt{2}(54r^2-6r)$.

4. 

$M_1\left({\gamma }_3\right)=270r^2-6r$.

5. 

$M_2\left({\gamma }_3\right)=324r^2-18r$.

6. 

$R{M}_2\left({\gamma }_3\right)=108r^2-12r$.

7. 

$HM\left({\gamma }_3\right)=1350r^2-54r$.

8. 

$AZ\left({\gamma }_3\right)=-{\displaystyle \frac{111}{4}}r+432r^2$.

9. 

$H\left({\gamma }_3\right)={\displaystyle \frac{3}{5}}r+{\displaystyle \frac{108}{5}}{r}^2$.

10. 

$SC\left({\gamma }_3\right)=3r+{\displaystyle \frac{\sqrt{5}(54r^2-6r)}{5}}$.

11. 

$GA\left({\gamma }_3\right)={\displaystyle \frac{6\sqrt{3}r(18\sqrt{10}r-2\sqrt{10}+5)}{5}}$.

12. 

$IS\left({\gamma }_3\right)=-{\displaystyle \frac{27}{10}}r+{\displaystyle \frac{324}{5}}{r}^2$.

13. 

$AZI\left({\gamma }_3\right)=117r^2+7r$.

14. 

$Alb\left({\gamma }_3\right)=54r^2+6r$.

15. 

$ABC\left({\gamma }_3\right)=\sqrt{3}\sqrt{9r^2-r}+2\sqrt{\mathrm{r}}$.

Proof. 

Let ${\gamma }_3$ be a hexagonal TriCF structure with $\mathcal{P}( )$ and $\mathcal{Q}( )$ as given in Table 3. We have conducted edge partitions of ${\gamma }_3$ based on the vertex degree, and the following results are obtained by applying those edge partitions in the definitions of degree-based topological indices (Table 1).

1. 

$R\left({\gamma }_3\right)={\displaystyle \frac{1}{\sqrt{1\times 3}}}\left(6r\right)+{\displaystyle \frac{1}{\sqrt{2\times 3}}}(54r^2-6r)$

$=2\sqrt{3}r+{\displaystyle \frac{\sqrt{6}(54r^2-6r)}{6}}$

$=\sqrt{3}r(9\sqrt{2}r-\sqrt{2}+2)$.

2. 

$RR\left({\gamma }_3\right)=\sqrt{1\times 3}\left(6r\right)+\sqrt{2\times 3}(54r^2-6r)$

$=6\sqrt{3}r+\sqrt{6}(54r^2-6r)$

$=6\sqrt{3}r(9\sqrt{2}r-\sqrt{2}+1)$.

3. 

$RRR\left({\gamma }_3\right)=\sqrt{(1-1)(3-1)}\left(6r\right)+\sqrt{(2-1)(3-1)}(54r^2-6r)$

$=\sqrt{2}(54r^2-6r)$.

4. 

$M_1\left({\gamma }_3\right)=(1+3)\left(6r\right)+(2+3)(54r^2-6r)$

$=270r^2-6r$.

5. 

$M_2\left({\gamma }_3\right)=(1\times 3)\left(6r\right)+(2\times 3)(54r^2-6r)$

$=324r^2-18r$.

6. 

$R{M}_2\left({\gamma }_3\right)=\left((1-1)(3-1)\right)\left(6r\right)+\left((2-1)(3-1)\right)(54r^2-6r)$

$=108r^2-12r$.

7. 

$HM\left({\gamma }_3\right)=\left({(1+3)}^2\right)\left(6r\right)+\left({(2+3)}^2\right)(54r^2-6r)$

$=1350r^2-54r$.

8. 

$AZ\left({\gamma }_3\right)={\left({\displaystyle \frac{1\times 3}{1+3-2}}\right)}^3\left(6r\right)+{\left({\displaystyle \frac{2\times 3}{2+3-2}}\right)}^3(54r^2-6r)$

$=-\frac{111}{4}r+432r^2$.

9. 

$H\left({\gamma }_3\right)={\displaystyle \frac{2}{1+3}}\left(6r\right)+{\displaystyle \frac{2}{2+3}}(54r^2-6r)$

$={\displaystyle \frac{3}{5}}r+{\displaystyle \frac{108}{5}}{r}^2$.

10. 

$SC\left({\gamma }_3\right)={\displaystyle \frac{1}{\sqrt{1+3}}}\left(6r\right)+{\displaystyle \frac{1}{\sqrt{2+3}}}(54r^2-6r)$

$=3r+{\displaystyle \frac{\sqrt{5}(54r^2-6r)}{5}}$.

11. 

$GA\left({\gamma }_3\right)=2\left({\displaystyle \frac{\sqrt{1\times 3}}{1+3}}\right)\left(6r\right)+2\left({\displaystyle \frac{\sqrt{2\times 3}}{2+3}}\right)(54r^2-6r)$

$=6\sqrt{3}r+{\displaystyle \frac{2\sqrt{30}(54r^2-6r)}{5}}$

$={\displaystyle \frac{6\sqrt{3}r(18\sqrt{10}r-2\sqrt{10}+5)}{5}}$.

12. 

$IS\left({\gamma }_3\right)=\left({\displaystyle \frac{1\times 3}{1+3}}\right)\left(6r\right)+\left({\displaystyle \frac{2\times 3}{2+3}}\right)(54r^2-6r)$

$=-{\displaystyle \frac{27}{10}}r+{\displaystyle \frac{324}{5}}{r}^2$.

13. 

$AZI\left({\gamma }_3\right)=\left({\displaystyle \frac{1+9}{1\times 3}}\right)\left(6r\right)+\left({\displaystyle \frac{4+9}{2\times 3}}\right)(54r^2-6r)$

$=117r^2+7r$.

14. 

$Alb\left({\gamma }_3\right)=2\left(6r\right)+1(54r^2-6r)$

$=54r^2+6r$.

15. 

$ABC\left({\gamma }_3\right)=\sqrt{{\displaystyle \frac{1+3-2}{1\times 3}}}\left(6r\right)+\sqrt{{\displaystyle \frac{2+3-2}{2\times 3}}}(54r^2-6r)$

$=2\sqrt{\mathrm{r}}+\sqrt{27r^2-3r}$

$=\sqrt{3}\sqrt{9r^2-r}+2\sqrt{\mathrm{r}}$.

□

4.2. Multiplicative Degree-Based Topological Indices of TriCF Structure

Theorem 4.

Let ${\gamma }_1$ be a linear chain TriCF structure with dimension r. Then, the degree-based multiplicative indices are

1. 

$R\left({\gamma }_1\right)={\left({\displaystyle \frac{\sqrt{3}}{3}}\right)}^{2r+4}{\left({\displaystyle \frac{\sqrt{6}}{6}}\right)}^{34r+14}$.

2. 

$RR\left({\gamma }_1\right)={\left(\sqrt{3}\right)}^{2r+4}{\left(\sqrt{6}\right)}^{34r+14}$.

3. 

$M_1\left({\gamma }_1\right)=4^{(2r+4)}$$5^{(34r+14)}$.

4. 

$M_2\left({\gamma }_1\right)=3^{2r+4}$$6^{34r+14}$.

5. 

$HM\left({\gamma }_1\right)={16}^{2r+4}$${25}^{34r+14}$.

6. 

$AZ\left({\gamma }_1\right)={\left({\displaystyle \frac{27}{8}}\right)}^{2r+4}$$8^{34r+14}$.

7. 

$H\left({\gamma }_1\right)={\left({\displaystyle \frac{1}{2}}\right)}^{2r+4}{\left({\displaystyle \frac{2}{5}}\right)}^{34r+14}$.

8. 

$SC\left({\gamma }_1\right)={\left({\displaystyle \frac{1}{2}}\right)}^{2r+4}{\left({\displaystyle \frac{\sqrt{5}}{5}}\right)}^{34r+14}$.

9. 

$GA\left({\gamma }_1\right)={\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^{2r+4}{\left({\displaystyle \frac{2\sqrt{6}}{5}}\right)}^{34r+14}$.

10. 

$IS\left({\gamma }_1\right)={\left({\displaystyle \frac{3}{4}}\right)}^{2r+4}{\left({\displaystyle \frac{6}{5}}\right)}^{34r+14}$.

11. 

$AZI\left({\gamma }_1\right)={\left({\displaystyle \frac{10}{3}}\right)}^{2r+4}{\left({\displaystyle \frac{13}{6}}\right)}^{34r+14}$.

12. 

$Alb\left({\gamma }_1\right)=2^{2r+4}$.

13. 

$ABC\left({\gamma }_1\right)={\left({\displaystyle \frac{\sqrt{6}}{3}}\right)}^{2r+4}{\left({\displaystyle \frac{\sqrt{2}}{2}}\right)}^{34r+14}$.

Proof. 

Let ${\gamma }_1$ be a linear chain TriCF structure with vertices $\mathcal{P}( )$ and $\mathcal{Q}( )$ (see Table 3). We have conducted edge partitions of ${\gamma }_1$ based on the vertex degree, and the following results are obtained by applying those edge partitions in the definitions of multiplicative degree-based topological indices (Table 2).

1. 

$R\left({\gamma }_1\right)={\left({\displaystyle \frac{1}{1\times 3}}\right)}^{(2r+4)}{\left({\displaystyle \frac{1}{2\times 3}}\right)}^{(34r+14)}$

$={\left({\displaystyle \frac{\sqrt{3}}{3}}\right)}^{2r+4}{\left({\displaystyle \frac{\sqrt{6}}{6}}\right)}^{34r+14}$.

2. 

$RR\left({\gamma }_1\right)={\left(\sqrt{1\times 3}\right)}^{(2r+4)}{\left(\sqrt{2\times 3}\right)}^{(34r+14)}$$={\left(\sqrt{3}\right)}^{2r+4}{\left(\sqrt{6}\right)}^{34r+14}$.

3. 

$M_1\left({\gamma }_1\right)={\left(1+3\right)}^{(2r+4)}{\left(2+3\right)}^{(34r+14)}$$=4^{(2r+4)}$$5^{(34r+14)}$.

4. 

$M_2\left({\gamma }_1\right)={\left(1\times 3\right)}^{(2r+4)}{\left(2\times 3\right)}^{(34r+14)}$$=3^{2r+4}$$6^{34r+14}$.

5. 

$HM\left({\gamma }_1\right)={\left({(1+3)}^2\right)}^{(2r+4)}{\left({(2+3)}^2\right)}^{(34r+14)}$

$={16}^{2r+4}$${25}^{34r+14}$.

6. 

$AZ\left({\gamma }_1\right)={\left({\left({\displaystyle \frac{1\times 3}{1+3-2}}\right)}^3\right)}^{(2r+4)}{\left({\left({\displaystyle \frac{2\times 3}{2+3-2}}\right)}^3\right)}^{(34r+14)}$

$={\left({\displaystyle \frac{27}{8}}\right)}^{2r+4}$$8^{34r+14}$.

7. 

$H\left({\gamma }_1\right)={\left({\displaystyle \frac{2}{1+3}}\right)}^{(2r+4)}{\left({\displaystyle \frac{2}{2+3}}\right)}^{(34r+14)}$

$={\left({\displaystyle \frac{1}{2}}\right)}^{2r+4}{\left({\displaystyle \frac{2}{5}}\right)}^{34r+14}$

8. 

$SC\left({\gamma }_1\right)={\left({\displaystyle \frac{1}{\sqrt{1+3}}}\right)}^{(2r+4)}{\left({\displaystyle \frac{1}{\sqrt{2+3}}}\right)}^{(34r+14)}$

$={\left({\displaystyle \frac{1}{2}}\right)}^{2r+4}{\left({\displaystyle \frac{\sqrt{5}}{5}}\right)}^{34r+14}$.

9. 

$GA\left({\gamma }_1\right)={\left(2\left({\displaystyle \frac{\sqrt{1\times 3}}{1+3}}\right)\right)}^{(2r+4)}{\left(2\left({\displaystyle \frac{\sqrt{2\times 3}}{2+3}}\right)\right)}^{(34r+14)}$

$={\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^{2r+4}{\left({\displaystyle \frac{2\sqrt{6}}{5}}\right)}^{34r+14}$.

10. 

$IS\left({\gamma }_1\right)={\left({\displaystyle \frac{1\times 3}{1+3}}\right)}^{(2r+4)}{\left({\displaystyle \frac{2\times 3}{2+3}}\right)}^{(34r+14)}$

$={\left({\displaystyle \frac{3}{4}}\right)}^{2r+4}{\left({\displaystyle \frac{6}{5}}\right)}^{34r+14}$.

11. 

$AZI\left({\gamma }_1\right)={\left({\displaystyle \frac{1+9}{1\times 3}}\right)}^{(2r+4)}{\left({\displaystyle \frac{4+9}{2\times 3}}\right)}^{(34r+14)}$

$={\left({\displaystyle \frac{10}{3}}\right)}^{2r+4}{\left({\displaystyle \frac{13}{6}}\right)}^{34r+14}$.

12. 

$Alb\left({\gamma }_1\right)={\left(2\right)}^{(2r+4)}{\left(1\right)}^{(34r+14)}$$=2^{2r+4}$.

13. 

$ABC\left({\gamma }_1\right)={\left(\sqrt{{\displaystyle \frac{1+3-2}{1\times 3}}}\right)}^{(2r+4)}{\left(\sqrt{{\displaystyle \frac{2+3-2}{2\times 3}}}\right)}^{(34r+14)}$

$={\left({\displaystyle \frac{\sqrt{6}}{3}}\right)}^{2r+4}{\left({\displaystyle \frac{\sqrt{2}}{2}}\right)}^{34r+14}$.

□

Theorem 5.

Let ${\gamma }_2$ be a parallelogram TriCF structure with dimensions r and s. Then, the multiplicative degree-based topological indices are

1. 

$R\left({\gamma }_2\right)={\left({\displaystyle \frac{\sqrt{3}}{3}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{\sqrt{6}}{6}}\right)}^{(18s+16)r+16s-2}$.

2. 

$RR\left({\gamma }_2\right)={\left(\sqrt{3}\right)}^{2r+2s+2}{\left(\sqrt{6}\right)}^{(18s+16)r+16s-2}$.

3. 

$M_1\left({\gamma }_2\right)=4^{2r+2s+2}$$5^{(18s+16)r+16s-2}$.

4. 

$M_2\left({\gamma }_2\right)=3^{2r+2s+2}$$6^{(18s+16)r+16s-2}$.

5. 

$HM\left({\gamma }_2\right)={16}^{2r+2s+2}$${25}^{(18s+16)r+16s-2}$.

6. 

$AZ\left({\gamma }_2\right)={\left({\displaystyle \frac{27}{8}}\right)}^{2r+2s+2}$$8^{(18s+16)r+16s-2}$.

7. 

$H\left({\gamma }_2\right)={\left({\displaystyle \frac{1}{2}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{2}{5}}\right)}^{(18s+16)r+16s-2}$.

8. 

$SC\left({\gamma }_2\right)={\left({\displaystyle \frac{1}{2}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{\sqrt{5}}{5}}\right)}^{(18s+16)r+16s-2}$.

9. 

$GA\left({\gamma }_2\right)={\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{2\sqrt{6}}{5}}\right)}^{(18s+16)r+16s-2}$.

10. 

$IS\left({\gamma }_2\right)={\left({\displaystyle \frac{3}{4}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{6}{5}}\right)}^{(18s+16)r+16s-2}$.

11. 

$Alb\left({\gamma }_2\right)=2^{2r+2s+2}$.

12. 

$AZI\left({\gamma }_2\right)={\left({\displaystyle \frac{10}{3}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{13}{6}}\right)}^{(18s+16)r+16s-2}$.

13. 

$ABC\left({\gamma }_2\right)={\left({\displaystyle \frac{\sqrt{6}}{3}}\right)}^{2r+2s+2}{\left(\frac{\sqrt{2}}{2}\right)}^{(18s+16)r+16s-2}$.

Proof. 

Let ${\gamma }_2$ be a parallelogram TriCF structure with vertex and edge set $\mathcal{P}$ and $\mathcal{Q}$, respectively (Table 3). We have conducted edge partitions of ${\gamma }_2$ based on the vertex degree (Table 3), and the following results are obtained by applying those edge partitions in the definitions of multiplicative degree-based topological indices (Table 2).

1. 

$R\left({\gamma }_2\right)={\left({\displaystyle \frac{1}{\sqrt{1\times 3}}}\right)}^{(2r+2s+2)}{\left({\displaystyle \frac{1}{\sqrt{2\times 3}}}\right)}^{(18s+16)r+16s-2}$

$={\left({\displaystyle \frac{\sqrt{3}}{3}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{\sqrt{6}}{6}}\right)}^{(18s+16)r+16s-2}$.

2. 

$RR\left({\gamma }_2\right)={\left(\sqrt{1\times 3}\right)}^{(2r+2s+2)}{\left(\sqrt{2\times 3}\right)}^{(18s+16)r+16s-2}$

$={\left(\sqrt{3}\right)}^{2r+2s+2}{\left(\sqrt{6}\right)}^{(18s+16)r+16s-2}$.

3. 

$M1\left({\gamma }_2\right)={\left(1+3\right)}^{(2r+2s+2)}{\left(2+3\right)}^{(18s+16)r+16s-2}$

$=4^{2r+2s+2}$$5^{(18s+16)r+16s-2}$.

4. 

$M2\left({\gamma }_2\right)={\left(1\times 3\right)}^{(2r+2s+2)}{\left(2\times 3\right)}^{(18s+16)r+16s-2}$

$=3^{2r+2s+2}$$6^{(18s+16)r+16s-2}$.

5. 

$HM\left({\gamma }_2\right)={\left({\left(1+3\right)}^2\right)}^{(2r+2s+2)}{\left({\left(2+3\right)}^2\right)}^{(18s+16)r+16s-2}$

$={16}^{2r+2s+2}$${25}^{(18s+16)r+16s-2}$

6. 

$AZ\left({\gamma }_2\right)={\left({\left({\displaystyle \frac{1\times 3}{(1+3-2)}}\right)}^3\right)}^{(2r+2s+2)}{\left({\left({\displaystyle \frac{2\times 3}{(2+3-2)}}\right)}^3\right)}^{(18s+16)r+16s-2}$

$={\left({\displaystyle \frac{27}{8}}\right)}^{2r+2s+2}$$8^{(18s+16)r+16s-2}$.

7. 

$H\left({\gamma }_2\right)={\left({\displaystyle \frac{2}{1+3}}\right)}^{(2r+2s+2)}{\left({\displaystyle \frac{2}{2+3}}\right)}^{(18s+16)r+16s-2}$

$={\left({\displaystyle \frac{1}{2}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{2}{5}}\right)}^{(18s+16)r+16s-2}$.

8. 

$SC\left({\gamma }_2\right)={\left({\displaystyle \frac{1}{\sqrt{1+3}}}\right)}^{(2r+2s+2)}{\left({\displaystyle \frac{2}{\sqrt{2+3}}}\right)}^{(18s+16)r+16s-2}$

$={\left({\displaystyle \frac{1}{2}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{\sqrt{5}}{5}}\right)}^{(18s+16)r+16s-2}$.

9. 

$GA\left({\gamma }_2\right)={\left(2\left({\displaystyle \frac{\sqrt{1\times 3}}{1+3}}\right)\right)}^{(2r+2s+2)}{\left(2\left({\displaystyle \frac{\sqrt{2\times 3}}{2+3}}\right)\right)}^{(18s+16)r+16s-2}$

$={\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{2\sqrt{6}}{5}}\right)}^{(18s+16)r+16s-2}$.

10. 

$IS\left({\gamma }_2\right)={\left({\displaystyle \frac{1\times 3}{1+3}}\right)}^{(2r+2s+2)}{\left({\displaystyle \frac{2\times 3}{2+3}}\right)}^{(18s+16)r+16s-2}$

$={\left({\displaystyle \frac{3}{4}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{6}{5}}\right)}^{(18s+16)r+16s-2}$.

11. 

$Alb\left({\gamma }_2\right)={\left(2\right)}^{(2r+2s+2)}{\left(1\right)}^{(18s+16)r+16s-2}$

$=2^{2r+2s+2}$.

12. 

$AZI\left({\gamma }_2\right)={\left({\displaystyle \frac{1+9}{1\times 3}}\right)}^{(2r+2s+2)}{\left({\displaystyle \frac{4+9}{2\times 3}}\right)}^{(18s+16)r+16s-2}$

$={\left({\displaystyle \frac{10}{3}}\right)}^{2r+2s+2}{\left({\displaystyle \frac{13}{6}}\right)}^{(18s+16)r+16s-2}$.

13. 

$ABC\left({\gamma }_2\right)={\left(\sqrt{{\displaystyle \frac{(1+3-2)}{(1\times 3)}}}\right)}^{(2r+2s+2)}{\left(\sqrt{{\displaystyle \frac{(2+3-2)}{(2\times 3)}}}\right)}^{(18s+16)r+16s-2}$

$={\left({\displaystyle \frac{\sqrt{6}}{3}}\right)}^{2r+2s+2}{\left(\frac{\sqrt{2}}{2}\right)}^{(18s+16)r+16s-2}$.

□

Theorem 6.

Let ${\gamma }_3$ be a hexagonal TriCF structure with dimension r. Then, the multiplicative degree-based topological indices are

1. 

$R\left({\gamma }_3\right)={\left({\displaystyle \frac{\sqrt{3}}{3}}\right)}^{6r}{\left({\displaystyle \frac{\sqrt{6}}{6}}\right)}^{54r^2-6r}$.

2. 

$RR\left({\gamma }_3\right)={\left(\sqrt{3}\right)}^{6r}{\left(\sqrt{6}\right)}^{54r^2-6r}$.

3. 

$M_1\left({\gamma }_3\right)=4^{6r}$$5^{54r^2-6r}$.

4. 

$M_2\left({\gamma }_3\right)=3^{6r}$$6^{54r^2-6r}$.

5. 

$HM\left({\gamma }_3\right)={16}^{6r}$${25}^{54r^2-6r}$.

6. 

$AZ\left({\gamma }_3\right)={\left({\displaystyle \frac{27}{8}}\right)}^{6r}$$8^{54r^2-6r}$.

7. 

$H\left({\gamma }_3\right)={\left({\displaystyle \frac{1}{2}}\right)}^{6r}{\left({\displaystyle \frac{2}{5}}\right)}^{54r^2-6r}$.

8. 

$SC\left({\gamma }_3\right)={\left({\displaystyle \frac{1}{2}}\right)}^{6r}{\left({\displaystyle \frac{\sqrt{5}}{5}}\right)}^{54r^2-6r}$.

9. 

$GA\left({\gamma }_3\right)={\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^{6r}{\left({\displaystyle \frac{2\sqrt{6}}{5}}\right)}^{54r^2-6r}$.

10. 

$IS\left({\gamma }_3\right)={\left({\displaystyle \frac{3}{4}}\right)}^{6r}{\left({\displaystyle \frac{6}{5}}\right)}^{54r^2-6r}$.

11. 

$AZI\left({\gamma }_3\right)=2340r^3-260r^2$.

12. 

$Alb\left({\gamma }_3\right)=2^{6r}$.

13. 

$ABC\left({\gamma }_3\right)={\left({\displaystyle \frac{\sqrt{6}}{3}}\right)}^{6r}{\left({\displaystyle \frac{\sqrt{2}}{2}}\right)}^{54r^2-6r}$.

Proof. 

Let ${\gamma }_3$ be a hexagonal TriCF structure. The cardinality of vertices $\mathcal{P}( )$ and edges $\mathcal{Q}( )$ are given in Table 3. We have conducted edge partitions of ${\gamma }_3$ based on the vertex degree, and the following results are obtained by applying those edge partitions in the definitions of multiplicative degree-based topological indices (Table 2)

1. 

$R\left({\gamma }_3\right)={\left({\displaystyle \frac{1}{\sqrt{1\times 3}}}\right)}^{6r}{\left({\displaystyle \frac{1}{\sqrt{1\times 3}}}\right)}^{54r^2-6r}$

$={\left({\displaystyle \frac{\sqrt{3}}{3}}\right)}^{6r}{\left({\displaystyle \frac{\sqrt{6}}{6}}\right)}^{54r^2-6r}$.

2. 

$RR\left({\gamma }_3\right)={\left(\sqrt{1\times 3}\right)}^{6r}{\left(\sqrt{2\times 3}\right)}^{54r^2-6r}$

$={\left(\sqrt{3}\right)}^{6r}{\left(\sqrt{6}\right)}^{54r^2-6r}$.

3. 

$M_1\left({\gamma }_3\right)={\left(1+3\right)}^{6r}{\left(2+3\right)}^{54r^2-6r}$

$=4^{6r}$$5^{54r^2-6r}$.

4. 

$M_2\left({\gamma }_3\right)={\left(1\times 3\right)}^{6r}{\left(2\times 3\right)}^{54r^2-6r}$

$=3^{6r}$$6^{54r^2-6r}$.

5. 

$HM\left({\gamma }_3\right)={\left({\left(1+3\right)}^2\right)}^{6r}{\left({\left(2+3\right)}^2\right)}^{54r^2-6r}$

$={16}^{6r}$${25}^{54r^2-6r}$.

6. 

$AZ\left({\gamma }_3\right)={\left({\left({\displaystyle \frac{1\times 3}{1+3-2}}\right)}^3\right)}^{6r}{\left({\left({\displaystyle \frac{1\times 3}{1+3-2}}\right)}^3\right)}^{54r^2-6r}$

$={\left({\displaystyle \frac{27}{8}}\right)}^{6r}$$8^{54r^2-6r}$.

7. 

$H\left({\gamma }_3\right)={\left({\displaystyle \frac{2}{1+3}}\right)}^{6r}{\left({\displaystyle \frac{2}{2+3}}\right)}^{54r^2-6r}$

$={\left(\frac{1}{2}\right)}^{6r}{\left(\frac{2}{5}\right)}^{54r^2-6r}$.

8. 

$SC\left({\gamma }_3\right)={\left({\displaystyle \frac{1}{\sqrt{1+3}}}\right)}^{6r}{\left({\displaystyle \frac{1}{\sqrt{2+3}}}\right)}^{54r^2-6r}$

$={\left({\displaystyle \frac{1}{2}}\right)}^{6r}{\left({\displaystyle \frac{\sqrt{5}}{5}}\right)}^{54r^2-6r}$.

9. 

$GA\left({\gamma }_3\right)={\left(2\left({\displaystyle \frac{\sqrt{1\times 3}}{1+3}}\right)\right)}^{6r}{\left(2\left({\displaystyle \frac{\sqrt{2\times 3}}{2+3}}\right)\right)}^{54r^2-6r}$

$={\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^{6r}{\left({\displaystyle \frac{2\sqrt{6}}{5}}\right)}^{54r^2-6r}$.

10. 

$IS\left({\gamma }_3\right)={\left({\displaystyle \frac{1\times 3}{1+3}}\right)}^{6r}{\left({\displaystyle \frac{2\times 3}{2+3}}\right)}^{54r^2-6r}$

$={\left({\displaystyle \frac{3}{4}}\right)}^{6r}{\left({\displaystyle \frac{6}{5}}\right)}^{54r^2-6r}$.

11. 

$AZI\left({\gamma }_3\right)={\left({\displaystyle \frac{1+9}{\sqrt{1\times 3}}}\right)}^{6r}{\left({\displaystyle \frac{4+9}{\sqrt{2\times 3}}}\right)}^{54r^2-6r}$

$={\left({\displaystyle \frac{10}{3}}\right)}^{6r}{\left({\displaystyle \frac{13}{6}}\right)}^{54r^2-6r}$.

12. 

$Alb\left({\gamma }_3\right)={\left(2\right)}^{6r}{\left(1\right)}^{54r^2-6r}$

$=2^{6r}$.

13. 

$ABC\left({\gamma }_3\right)={\left(\sqrt{{\displaystyle \frac{(1+3-2)}{(1\times 3)}}}\right)}^{6r}{\left(\sqrt{{\displaystyle \frac{(2+3-2)}{(2\times 3)}}}\right)}^{54r^2-6r}$

$={\left({\displaystyle \frac{\sqrt{6}}{3}}\right)}^{6r}{\left({\displaystyle \frac{\sqrt{2}}{2}}\right)}^{54r^2-6r}$.

□

4.3. Degree-Based Entropy Measures

This section explains how to calculate entropy values using Shannon’s method by creating the probability function using degree-based topological indices. To calculate probabilistic entropy, we utilised Shannon’s model because it is the most popular approach [7,11,13,31]. The entropy measured using that topological index X is given by

${\displaystyle E_X\left(\gamma \right)=log\left(X\left(\gamma \right)\right)-{\displaystyle \frac{1}{X\left(\gamma \right)}}\left(\sum _{uv\in E\left(\gamma \right)}^{}f\left(e\right)log\left(f\left(e\right)\right)\right)}$

By using the first Zagreb index to calculate the entropy value for the TriCF structure, the calculation procedure is illustrated.

First Zagreb entropy for linear chain TriCF molecular graph,

${\displaystyle E{M}_1\left({\gamma }_1\right)=log\left(M_{1{\gamma }_1}\right)-{\displaystyle \frac{1}{M_{1{\gamma }_1}}}\sum _{𝓊𝓋\in \mathcal{Q}( )}\left(𝒹ℊ\left(𝓊\right)+𝒹ℊ\left(𝓋\right)\right)log(𝒹ℊ\left(𝓊\right)+𝒹ℊ\left(𝓋\right))}$

$=log(178r+86)-{\displaystyle \frac{1}{178r+86}}\left(\left((2r+4)(1+3)log(1+3)\right)+(34r+14)(2+3)log(2+3)\right)$

$log(178r+86)-{\displaystyle \frac{8(2r+4)log\left(2\right)+5(34r+14)log\left(5\right)}{178r+86}}$

After simplifying this, we obtain

$E{M}_1\left({\gamma }_1\right)={\displaystyle \frac{(89r+43)log(89r+43)+(81r+27)log\left(2\right)+(-85r-35)log\left(5\right)}{89r+43}}$.

The general entropy expression of each TriCF chemical network would be too long to provide as theorems. By using the above mentioned procedure, it is simple to construct any degree-based entropies expression with regard to each topological index.

5. Numerical Computation

The numerical results of degree- based topological descriptors utilizing entopy measures generated for three different TriCF structures are shown in this section, with the values of the variable r & s ranging from 1 to 10. For a graphical comparison of the computed topological descriptors, these values were plotted using the Orgin 2020 b software.

Table 4. Numerical values for degree-based entropies of linear chain TriCF structure.

r	$\mathit{ER}\left({\mathit{\gamma }}_1\right)$	$\mathit{ERR}\left({\mathit{\gamma }}_1\right)$	${\mathit{EM}}_1\left({\mathit{\gamma }}_1\right)$	${\mathit{EM}}_2\left({\mathit{\gamma }}_1\right)$	$\mathit{EHM}\left({\mathit{\gamma }}_1\right)$	$\mathit{EAZ}\left({\mathit{\gamma }}_1\right)$	$\mathit{EH}\left({\mathit{\gamma }}_1\right)$	$\mathit{ESC}\left({\mathit{\gamma }}_1\right)$	$\mathit{EGA}\left({\mathit{\gamma }}_1\right)$	$\mathit{EIS}\left({\mathit{\gamma }}_1\right)$	$\mathit{EAZI}\left({\mathit{\gamma }}_1\right)$	$\mathit{EAlb}\left({\mathit{\gamma }}_1\right)$
1	4.5044	5.6303	3.9868	3.9726	3.9812	3.9658	3.9862	3.9883	3.9866	3.9805	3.9776	3.9557
2	5.2513	6.1719	4.498	4.4866	4.4935	4.4812	4.4975	4.4993	4.4978	4.4929	4.4903	4.4718
3	5.7778	6.5215	4.8347	4.8244	4.8306	4.8196	4.8342	4.8358	4.8345	4.8301	4.8277	4.8107
4	6.1886	6.78	5.0861	5.0765	5.0823	5.072	5.0857	5.0871	5.0859	5.0818	5.0795	5.0634
5	6.527	6.9853	5.2868	5.2777	5.2832	5.2734	5.2864	5.2878	5.2867	5.2827	5.2805	5.265
6	6.8151	7.1555	5.4539	5.4451	5.4504	5.4409	5.4535	5.4549	5.4538	5.45	5.4478	5.4327
7	7.0662	7.301	5.5971	5.5884	5.5936	5.5844	5.5967	5.598	5.5969	5.5932	5.591	5.5763
8	7.2888	7.4279	5.7222	5.7138	5.7189	5.7098	5.7219	5.7232	5.7221	5.7185	5.7163	5.7018
9	7.4889	7.5406	5.8335	5.8252	5.8302	5.8212	5.8331	5.8344	5.8334	5.8298	5.8277	5.8134
10	7.6706	7.6418	5.9336	5.9254	5.9303	5.9215	5.9332	5.9345	5.9335	5.9299	5.9278	5.9137

,

Table 5. Numerical values for degree-based entropies of parallelogram TriCF structures.

$(\mathit{r},\mathit{s})$	$\mathit{ER}\left({\mathit{\gamma }}_2\right)$	$\mathit{ERR}\left({\mathit{\gamma }}_2\right)$	${\mathit{EM}}_1\left({\mathit{\gamma }}_2\right)$	${\mathit{EM}}_2\left({\mathit{\gamma }}_2\right)$	$\mathit{EHM}\left({\mathit{\gamma }}_2\right)$	$\mathit{EAZ}\left({\mathit{\gamma }}_2\right)$	$\mathit{EH}\left({\mathit{\gamma }}_2\right)$	$\mathit{ESC}\left({\mathit{\gamma }}_2\right)$	$\mathit{EGA}\left({\mathit{\gamma }}_2\right)$	$\mathit{EIS}\left({\mathit{\gamma }}_2\right)$	$\mathit{EAZI}\left({\mathit{\gamma }}_2\right)$	$\mathit{EAlb}\left({\mathit{\gamma }}_2\right)$
1	3.174	4.8518	5.5761	5.7236	7.167	6.002	3.1002	3.1974	4.7498	4.1288	4.8203	4.0944
2	4.1165	5.8451	6.5653	6.7262	8.1634	7.0083	4.0707	4.1733	5.7395	5.1257	5.7797	5.037
3	4.7315	6.479	7.1974	7.3639	8.7985	7.6474	4.695	4.7998	6.3718	5.7611	6.3991	5.649
4	5.1945	6.952	7.6695	7.8391	9.2722	8.1235	5.1625	5.2686	6.844	6.235	6.8638	6.1092
5	5.5676	7.3313	8.0482	8.2199	9.6519	8.5048	5.5381	5.6451	7.2227	6.6149	7.2375	6.48
6	5.8805	7.6486	8.365	8.5382	9.9696	8.8235	5.8528	5.9604	7.5396	6.9325	7.5508	6.7912
7	6.1504	7.9216	8.6376	8.812	10.2428	9.0975	6.1238	6.2319	7.8123	7.2059	7.8208	7.0596
8	6.3877	8.1613	8.8771	9.0523	10.4827	9.3381	6.362	6.4705	8.0517	7.4458	8.0582	7.2957
9	6.5996	8.3751	9.0907	9.2665	10.6967	9.5525	6.5745	6.6833	8.2653	7.6598	8.2701	7.5066
10	6.791	8.568	9.2834	9.4599	10.8897	9.746	6.7664	6.8754	8.4581	7.8529	8.4615	7.6971

and

Table 6. Numerical values for degree-based entropies of hexagonal TriCF structures.

r	$\mathit{ER}\left({\mathit{\gamma }}_3\right)$	$\mathit{ERR}\left({\mathit{\gamma }}_3\right)$	${\mathit{EM}}_1\left({\mathit{\gamma }}_3\right)$	${\mathit{EM}}_2\left({\mathit{\gamma }}_3\right)$	$\mathit{EHM}\left({\mathit{\gamma }}_3\right)$	$\mathit{EAZ}\left({\mathit{\gamma }}_3\right)$	$\mathit{EH}\left({\mathit{\gamma }}_3\right)$	$\mathit{ESC}\left({\mathit{\gamma }}_3\right)$	$\mathit{EGA}\left({\mathit{\gamma }}_3\right)$	$\mathit{EIS}\left({\mathit{\gamma }}_3\right)$	$\mathit{EAZI}\left({\mathit{\gamma }}_3\right)$	$\mathit{EAlb}\left({\mathit{\gamma }}_3\right)$
1	3.9819	3.984	3.9868	3.9726	3.9812	3.9658	3.9862	3.9883	3.9866	3.9805	3.9776	3.9557
2	5.3714	5.3727	5.3741	5.3669	5.3713	5.3635	5.3738	5.3749	5.374	5.3709	5.369	5.3564
3	6.1836	6.1845	6.1854	6.1806	6.1835	6.1783	6.1852	6.186	6.1854	6.1833	6.1819	6.1731
4	6.7596	6.7603	6.761	6.7573	6.7595	6.7557	6.7608	6.7614	6.7609	6.7593	6.7583	6.7515
5	7.2062	7.2068	7.2074	7.2045	7.2062	7.2031	7.2072	7.2077	7.2073	7.2061	7.2052	7.1997
6	7.5711	7.5716	7.5721	7.5697	7.5711	7.5686	7.572	7.5724	7.5721	7.571	7.5703	7.5656
7	7.8796	7.8801	7.8805	7.8784	7.8796	7.8774	7.8804	7.8807	7.8804	7.8795	7.8789	7.8749
8	7.8796	8.1472	8.1476	8.1457	8.1468	8.1449	8.1475	8.1478	8.1475	8.1467	8.1462	8.1427
9	8.3825	8.3828	8.1476	8.3815	8.3825	8.3808	8.3831	8.3834	8.3831	8.3824	8.3819	8.3788
10	8.5933	8.5936	8.5939	8.5925	8.5933	8.5918	8.5938	8.5941	8.5939	8.5933	8.5928	8.59

summarize the results. Figure 5 shows this trend as three-dimensional graphical repersentations. The difference between each topological index for a specific structure can be seen in these 3D plots. The 3D plots can also be used to compare the behavior of a given index for the three different structures that are being researched in this article. The following tables and figures examine various entropies in numerical and graphical forms for all the different structures of the TriCF molecular graph.

6. Conclusions

In this study, degree-based topological indices have been computed using the multiplicative and entropy measures. These indices allow chemists to predict a variety of molecular compound qualities without the need for costly or time-consuming tests. For instance, from Figure 6, reverse Randić has a greater impact on the initial structure than other indices if we analyse the entropy measures. As a result, the computed results may be extremely important in forecasting TriCF system features. This is a novel synthesized lubricant, and so far, no physicochemical properties have been experimentally studied. Therefore, this current study will help the researchers to make further progress. We have also computed the topological indices for this TriCF structure for future utilization. This paper also includes a graphical depiction and a numerical comparison of the computed results, which is helpful for both theoretical chemists and professionals in the field.