Analysis of Zigzag and Rhombic Benzenoid Systems via Irregularity Indices

Topological indices are numerical quantities associated with the molecular graph of a chemical structure. Tese indices are used to predict various properties of chemical structures. Imbalance-based analysis is an advanced technique used for chemical compounds with irregular characteristics. Te molecular graphs of zigzag benzenoid systems ( Z p ) and rhombic benzenoid systems ( R p ) are inherently irregular. Terefore, applying the imbalance technique to these molecular structures plays an important role in predicting diferent properties. In this paper, we calculate sixteen irregularity indices for both Z p and R p systems. By examining these indices, we aim to provide insights into the properties of these structures and ultimately contribute to a deeper understanding of the feld.


Introduction
Graph theory is a branch of mathematics that deals with the study of graphs and their properties.Graphs are a type of mathematical structure that consists of nodes (also called vertices) and edges, which represent connections between the nodes.Graph theory has many applications, including computer science, biology, social network analysis, and more.One of the fundamental concepts in graph theory is the idea of a path, which is a sequence of edges connecting two nodes (see [1,2]).Another important concept is the degree of a node, which refers to the number of edges connected to that node.Graph theory also includes the study of graph coloring, which involves assigning colors to the nodes of a graph in such a way that adjacent nodes have diferent colors.Te study of graph theory has led to many important results and discoveries and continues to be an active area of research (see [3]).
In this paper, all graphs under consideration are fnite, simple, and undirected.Let G be a graph with vertex set V and edge set E. Te degree of a vertex u in G is denoted by d u and is defned as the number of edges incident to vertex u.In other words, d u is the number of vertices adjacent to vertex u in G. Te distance between any two vertices in G is defned as the length of the shortest path between them.Terefore, the degree of a vertex u can also be defned as the number of vertices at distance one from u. Tis notion of degree is fundamental to the study of graphs and plays a key role in many of their properties and applications.
In mathematical chemistry, the study of the graphical structure of chemical compounds allows researchers to predict important properties without performing experiments in a laboratory.Tis feld of study is known as chemical graph theory [4,5].A simple graph is defned as a graph without loops or multiple edges.In chemical graph theory, atoms are represented by vertices (or nodes) and chemical bonds are represented by edges (or lines).Te degree of a vertex is defned as the number of edges that are incident to it.Te study of topological indices in chemical graph theory began with the Wiener index in 1947, followed by the Zagreb indices, which were introduced due to the vast number of applications of the Wiener index [6].Te mathematical formulas for the frst and second Zagreb indices are given by (1) In addition to the Wiener and Zagreb indices, many other types of indices have been introduced to analyze the topology of diferent chemical structures [7,8].To measure the irregularities present in a chemical structure, irregularity indices have been introduced [9,10].Tese indices provide valuable insights into the properties of chemical structures and can help researchers make predictions and discoveries in the feld of mathematical chemistry.
An irregularity measure index is a mathematical index that takes a value of either zero or greater than zero and is used to measure the irregularities present in a chemical structure [11].Tese indices are calculated based on the degree of vertices, which is a fundamental property of the chemical graph.Table 1 presents the mathematical formulas for these irregularity measure indices, which can be used to predict various properties of the chemical compounds under consideration.However, the interpretation and analysis of these indices can be complex, and their application requires a deep understanding of the underlying principles of chemical graph theory.Nonetheless, they provide valuable insights into the properties and behavior of the molecules and can aid in the discovery of new compounds with specifc properties.Terefore, the study of irregularity measure indices is crucial for researchers in the feld of mathematical chemistry.
For a detailed background on these irregularity measure indices, we refer readers to the literature [13][14][15][16].Tese studies provide a comprehensive overview of the theory and application of irregularity indices in chemical graph theory and cover a wide range of topics such as their mathematical properties, algorithmic complexity, and relationship with other topological indices.Te works also provide examples of their application in predicting various properties of chemical compounds, such as boiling points, melting points, and refractive indices.Terefore, a thorough understanding of these studies is essential for researchers interested in exploring the potential of irregularity indices in the feld of mathematical chemistry.
Benzenoid systems have signifcant importance in theoretical chemistry due to their natural graph representation of benzenoid hydrocarbons [17].In a hexagonal system, there exists a vertex that belongs to three hexagons, which is known as the internal vertex of the hexagonal system [18].Benzenoid hydrocarbons are commonly found in our surroundings, minerals, and food and are also produced as byproducts in certain reactions, with a wide range of applications in chemical synthesis [19].However, despite their widespread use, benzenoid hydrocarbons are known to be pollutants and carcinogenic.Benzenoid systems are essentially hydrogen-deprived benzenoid hydrocarbons [20].
Te authors in [21] emphasized the importance of the three-dimensional distribution of benzene.More information about this repetitive surface can be found in [22,23].Te structure needed to be connected as a threedimensional solid carbon; however, to our knowledge, no such sequence has been considered for such a purpose.Tis goal was aimed at raising the awareness of researchers towards the atomic recognition of such friendly concepts in carbon nanoscience [24,25].
Te aim of this present work is to comprehensively study the topological properties of zigzag and rhombic benzenoid systems, which are two important families of benzenoid systems.To achieve this goal, we start by sketching the graphs of these systems and determining the number of vertices and edges in each graph.We then classify the edges into diferent classes based on the degrees of the end vertices.Using these classifcations, we calculate 16 irregularity measures for each system.Our fndings provide valuable insights into the topological behavior of these systems.

Irregularity Indices for Zigzag Benzenoid System Z p .
Figure 1 shows the graph Z p , which consists of p rows, with each row consisting of two hexagonal units sharing one common edge.Te frst row of Z p contains eleven edges, while the second row has twenty-one edges.Continuing in the same pattern, we can deduce that Z p has 10p + 1 edges and 8p + 2 vertices.Tere are three types of edges present in Z p , namely, (2, 2), (2,3), and (3,3).We can partition the edges into three sets P 1 , P 2 , and P 3 , where P 1 has 2p + 4 edges, P 2 has 4p edges, and P 3 has 4p − 3 edges.Tese edge partitions are denoted as P � P 1 + P 2 + P 3 .Tese observations provide useful information about the structure of Zp and can aid researchers in understanding the topological properties of this important family of benzenoid systems.Table 2 displays the three edge partitions P1, P2, and P3 of the hexagonal system Z p .

Theorem 1. Consider the graph
Journal of Mathematics Proof.We can obtain our desired results by utilizing the mathematical formulas for irregularity indices given in Table 1 and the edge partitions of Z p presented in Table 2.

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Irregularity Indices for Rhombic Benzenoid System R p .
Te graph of R p is shown in Figure 2. We can see that R p contains three types of edges, namely, (2, 2), (2, 3), and (3,3).Te edge set of R p can be partitioned into three      It can be shown that the total number of edges in R p is 32pq − 2p − 2q.Tis result can be obtained by counting the number of edges in each row of the graph and summing over all rows.Specifcally, the number of edges in the ith row is 8(p + q − i) + 4, for 1 ≤ i ≤ p + q − 1, and the number of edges in the last row is 4p + 4q − 4. Summing over all rows, we obtain the total number of edges as 32pq − 2p − 2q,      which can be verifed by direct calculation.Table 3 presents the edge partition of R p .Te frst column shows the types of edges, i.e., (2, 2), (2,3), and (3,3), and the second column shows the number of edges of each type.
Theorem .Consider the graph G corresponding to the rhombic benzenoid system R p .Ten, we have      Journal of Mathematics Proof.Using the mathematical formulas of irregularity indices given in Table 1 and the edge partition of the rhombic benzenoid system R p given in Table 3, we can perform the following computations to obtain our desired results.

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Discussion and Graphical Representation
In this section, we present a graphical comparison of both benzenoid systems Z p and R p .Te color red is fxed for the Z p graph, while the color green is used for the R p graph.Tis visual representation provides a clear comparison between the two systems, highlighting their similarities and diferences.
From Figures 3-18, we can see that by fxing the values of structural parameters involved, we can control the irregularities of both Z p and R p .Tis is an important observation as it allows us to tailor the properties of these systems to suit specifc applications.Furthermore, the graphical comparison in Figure 3 highlights the fact that diferent irregularities behave diferently.Some irregularities in Z p increase faster than those in R p , while others do not increase faster.Tis is important because it suggests that the behavior of a particular irregularity is dependent on the specifc parameters of the system under study.By carefully choosing these parameters, we can optimize the behavior of the system with respect to a particular irregularity.Overall, the results presented in this section provide valuable insights into the behavior of benzenoid systems and may prove useful in the design of materials for various applications.

Conclusion
In this study, we have presented a comprehensive analysis of the irregularity indices of Z p and R p , which are two structurally complex systems.Our fndings reveal that these systems exhibit a high degree of irregularity, and the calculated indices provide valuable insight into their nature.Te obtained results can be applied in the quantitative structure-activity relationship modeling of various physical and chemical properties of these systems.Additionally, the graphical comparison between Z p and R p highlights the diferences in their irregularity behaviors, which can aid in the further understanding and characterization of these structures.Overall, this work contributes to the growing body of literature on irregularity indices and their applications in chemical graph theory.

Future Directions
Here are some possible future directions based on the fndings of this study: (1) Investigating the relationship between irregularity indices and physical/chemical properties of Z p and R p : this can be done through quantitative structureactivity relationship (QSAR) modeling, which can help predict properties of these structures and guide the design of new molecules with desirable properties.(2) Studying the efect of varying the structural parameters of Z p and R p on their irregularity indices: this can provide insights into the factors that contribute to the irregular nature of these structures and guide the design of new structures with desired irregularity properties.(3) Exploring the application of irregularity indices in other types of networks, such as biological networks and social networks: the use of irregularity indices can help characterize the complex structure of these networks and guide the design of more efcient and robust systems.(4) Developing new methods for calculating irregularity indices that can handle larger and more complex networks: this can help extend the application of irregularity indices to a wider range of structures and facilitate the study of their properties.

Limitations of the Used Method
(1) Te method is limited to calculating only the sixteen irregularity indices and may not be applicable to other types of indices or measures of irregularity.(2) Te method relies on the assumption that the considered graphs have an irregular nature, which may not be true for all types of graphs.(3) Te method assumes that the structural parameters involved are the only factors afecting the irregularity Journal of Mathematics indices, while other factors, such as environmental conditions, may also play a role in determining the properties of the underlying structures.(4) Te method does not consider the dynamic nature of the underlying structures and may not be suitable for analyzing their behavior over time.(5) Te method may have limitations in its applicability to real-world systems, as the structures considered in the study are theoretical models and may not fully refect the complexity of real-world systems.

Robustness of the Proposed Method
Te proposed method for calculating irregularity indices can be easily applied to other graph structures beyond Z p and R p .Terefore, the method shows promising robustness and versatility in analyzing the irregularities of diferent types of graphs.

Figure 1 :
Figure 1: Molecular graph of Z p .

Figure 2 :
Figure 2: Molecular graph of R p .