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].
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,
,
and
are linear chain TriCF structures, parellelogram TriCF structures and hexagonal TriCF structures, respectively.
The total number of vertices and edges of is depicted as = and = , respectively. In addition, the edge set has two edge partitions. The first edge partition contains edges with the following values: ; = 1, = 3 and edges make up the second edge partition, where = 2, = 3.
The total number of vertices and edges of is depicted as = and = , respectively. In addition, the edge set has two edge partitions. The first edge partition contains edges with the following values: ; = 1, = 3 and edges make up the second edge partition, where = 2,= 3.
The total number of vertices and edges of is depicted as = and = , respectively. In addition, the edge set has two edge partitions. The first edge partition contains edges with the following values: ; = 1, = 3 and edges make up the second edge partition, where = 2, = 3.
Table 3 below illustrates how the structure’s edges are divided.
4.1. Degree-Based Topological Indices of TriCF Structure
Theorem 1. Let be a linear chain TriCF structure with dimension r. Then,
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Proof. Let
be a linear chain TriCF structure with
and
(see
Table 3). We have conducted edge partitions of
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).
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Theorem 2. Let be a parallelogram TriCF structure with dimensions r and s. Then, the degree-based topological indices are:
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Proof. Let
be a parallelogram TriCF structure with
and
as taken from
Table 3. We have conducted edge partitions of
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).
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Theorem 3. Let be a hexagonal TriCF structure with dimension r. Then,
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Proof. Let
be a hexagonal TriCF structure with
and
as given in
Table 3. We have conducted edge partitions of
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).
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4.2. Multiplicative Degree-Based Topological Indices of TriCF Structure
Theorem 4. Let be a linear chain TriCF structure with dimension r. Then, the degree-based multiplicative indices are
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Proof. Let
be a linear chain TriCF structure with vertices
and
(see
Table 3). We have conducted edge partitions of
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).
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Theorem 5. Let be a parallelogram TriCF structure with dimensions r and s. Then, the multiplicative degree-based topological indices are
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Proof. Let
be a parallelogram TriCF structure with vertex and edge set
and
, respectively (
Table 3). We have conducted edge partitions of
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).
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Theorem 6. Let be a hexagonal TriCF structure with dimension r. Then, the multiplicative degree-based topological indices are
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Proof. Let
be a hexagonal TriCF structure. The cardinality of vertices
and edges
are given in
Table 3. We have conducted edge partitions of
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)
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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
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,
After simplifying this, we obtain
.
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.