Abstract
The molecular topology and the chemical structures which are not regular perform a dominant character in designing the compound in connection with their physical and chemical properties. The study of topological descriptors of molecular structures is widely used in the field of cheminformatics, information technology, biomedical science, and many more. Numerous topological indices have been evolved in the theory of chemistry. Graphs which are not regular can be specified using irregular measures. In this article, we find some irregularity indices, which are useful in Quantitative structure activity relationship for benzenoid structures. Based on the exact analytic expressions, the numerical and graphical comparison for benzenoid structures is also provided.
1. Introduction
In chemical graph theory, latest innovation in graph theoretical models and simulation of molecular graphs are conducted by various researchers. They empower the researchers to develop a correlation between graph theory and chemical compounds. Also, the researches explain the physical and chemical properties like freezing point, solubility, and melting point of chemical structure of various compounds [1–3]. An irregularity index is an analytical value associated with a chemical structure that describes a structure’s irregularity. Von Collatz and Ulrich defined the notion “network irregularity” in 1957 [4]. Irregularity measures are aided to denote various organizations in convoluted networks. Self-similarity, network motif, and scale-freeness are all main topological features of these organizations [5].
Let be an ordered pair of graphs, where is a nonempty vertex set and is an edge set. Topological descriptors can be obtained from the molecular structure. They are very helpful to explain Quantitative Structure Activity Relationship (QSAR) of medicines and chemical compounds to form their molecular characteristics by computing numerically [6].
The total number of edges incident with vertex is called the degree of vertex , it is denoted by . In a graph, the maximum vertex degree is denoted by . Topological study of Line graph of Remdesivir Compound used in the treatment of corona virus is discussed [7]. Degree-based topological indices and polynomials of hyaluronic acid-curcumin conjugates, locating and Multiplicative Locating Indices of Graphs with QSPR Analysis, and structural determination of Paraffin Boiling Points are well studied [3, 8–12]. Graph Irregularity Indices used as Molecular Descriptors in QSPR studies and Randic Index, Irregularity, and complex biomolecular networks are also extensively discussed [13, 14]. The expected values of arithmetic bond connectivity and geometric indices in random phenylene chains are clearly presented [15–22]. In literature, Irregularity indices of metal organic frameworks [23], Bismuth Tri-Iodide [24], probabilistic neural network [25], and computer networks like silicate, oxide, hexagonal, and honeycomb [26] were extensively studied. Gutman studied the topological indices and irregularity measures [27]. Graphs with equal irregularity indices are well explained [28]. Irregularity indices based on Zagreb indices were studied [29]. For line graph of Dutch windmill graph, the irregularity indices were extensively studied [30]. Several irregularity indices of Dendrimers structures were clearly discussed [31].
In this article, we compute the irregularity degree based topological indices of the benzenoid planar octahedron networks and compare the results graphically.
In 1972, Gutman discussed the first and second Zagreb indices [32].
In a graph, if all the degrees are same, then the graph is called a regular graph. Otherwise, it is irregular. The selected irregularity indices for benzenoid planar octahedron networks are represented by
2. Main Outcomes
The structure of Benzenoid planar octahedron is constructed and the topological properties of this structure is well discussed [33]. The 2-dimensional structure of Benzenoid planar octahedron structure is given in Figure 1. We represent the graph of Benzenoid planar octahedron by , , where n denotes the dimension of Benzenoid planar octahedron. Figure 1 depicts the 2-dimensional structure of the molecular graph of . The total number of vertices (q) and edges (p) in the structure of Benzenoid planar octahedron are and .
The set of edges of Benzenoid planar octahedron structure is divided into five partitions based on the degree of end vertices. The first edge partition includes of edges , where and . The second edge partition includes of edges , where and . The third edge partition includes of edges , where and . The fourth edge partition includes of edges , where and . The fifth edge partition includes of edges , where and . Table 1 shows the edge partition of Benzenoid planar octahedron structure.
Now the irregularity degree based topological indices is computed as.
Theorem 1. We consider the graph , , then the irregularity indices of Benzenoid planar octahedron structure are
Proof. Applying the partition of edges of end vertices of each edge of the Benzenoid planar octahedron structure exploited in Table 1, we calculate the irregularity indices of the Benzenoid planar octahedron structure and the mathematical computations are given as follows:The structure of Benzenoid dominating planar octahedron is constructed and the topological properties of this structure is well explained [33]. The 2-dimensional structure of Benzenoid dominating planar octahedron structure is given in Figure 2. We denote the graph of Benzenoid dominating planar octahedron by , , where n denotes the dimension of Benzenoid dominating planar octahedron. Figure 2 depicts the 2-dimensional structure of the molecular graph of . The total number of vertices (t) and edges (s) in the structure of Benzenoid dominating planar octahedron are and .
The set of edges of Benzenoid dominating planar octahedron structure is divided into five partitions based on the degree of end vertices. The first partition of edge includes of edges , where and . The second partition of edge includes of edges , where and . The third partition of edge includes of edges , where and . The fourth partition of edge includes of edges , where and . The fifth partition of edge includes of edges , where and . Table 2 shows the edge partition of Benzenoid dominating planar octahedron structure.
Theorem 2. Consider the graph , , then the irregularity indices of Benzenoid dominating planar octahedron structure are
Proof. Applying the partition of edge based on degrees of end vertices of each edge of the Benzenoid dominating planar octahedron structure depicted in Table 2, we compute the irregularity indices of the Benzenoid dominating planar octahedron structure , and the mathematical calculations are given as follows:The structure of Benzenoid hex planar octahedron is constructed and the topological properties of this structure are well discussed [33]. The 2-dimensional structure of Benzenoid hex planar octahedron structure is given in Figure 3. We represent the graph of Benzenoid hex planar octahedron by , , where n denotes the dimension of Benzenoid hex planar octahedron. Figure 3 depicts the 2-dimensional structure of the molecular graph of . The total number of vertices (u) and edges (v) in the structure of Benzenoid hex planar octahedron are and .
The set of edges of Benzenoid hex planar octahedron structure is divided into five partitions based on the degree of end vertices. The first partition of edge includes of edges , where and . The second partition of edge includes of edges , where and . The third partition of edge includes of edges , where and . The fourth partition of edge includes of edges , where and . The fifth partition of edge includes of edges , where and . The sixth partition of edge includes of edges , where and . The seventh partition of edge includes of edges , where and . Table 3 shows the edge partition of Benzenoid hex planar octahedron structure.
Theorem 3. Consider the graph , , then the irregularity indices of Benzenoid hex planar octahedron structure are
Proof. Applying the partition of edge based on degrees of end vertices of each edge of the Benzenoid hex planar octahedron structure depicted in Table 3, we compute the irregularity indices of the Benzenoid hex planar octahedron structure and the mathematical calculations are given as follows:
3. Graphical Representation of the Chemical Structures
Here, we differentiate the outcomes of irregularity indices of chemical structures graphically. Three different colours have been exploited to show the graphical behaviour of irregularity indices for chemical structures. These graphs have been existed by substituting the numerical values of n through the X-axis with respect to the outcomes of irregularity indices through the Y-axis. Tables 4–6 depict such numerical values of outcomes, which can be existed after substituting the numerical values of n, and these numerical values exhibited in the tables can aid us to make graphical outcomes. Graphical lines with three distinct colours have been employed from Figure 4. Blue colour denotes the behaviour of irregular indices in the Benzenoid planar octahedron structure, red colour denotes the behaviour of irregular indices in the Benzenoid dominating planar octahedron structure, and green colour denotes the behaviour of irregular indices in the Benzenoid hex planar octahedron structure. Numerical values of n are plotted through the X-axis and two dimensional graphical behaviour of the irregularity indices through the Y-axis are shown in Figure 4. The three dimensional graphical behaviour of the irregularity indices of benzenoid structures are shown in Figures 5–7.
4. Conclusion
In the analysis of the quantitative structure property relationships (QSPRs) and (QSARs), chemical indices are important tools to approximate the characteristics of the bioactivity, physical, biomedicine, and chemical compounds. In this paper, we have provided the results on irregularity chemical indices as depicted in Figures 4–7 for Benzenoid planar octahedron structure, Benzenoid dominating planar octahedron structure, and Benzenoid hex planar octahedron structure, besides indices showed increased values for Benzenoid planar octahedron structure, Benzenoid dominating planar octahedron structure, and Benzenoid hex planar octahedron structure. The computational results which we get will aid the investigators to recognise the preferred structure more easily and would inspire the others to concentrate on the Benzenoid planar octahedron structure, Benzenoid dominating planar octahedron structure, and Benzenoid hex planar octahedron structure.
Data Availability
All the data and material used in this research is included in the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.