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Irregularity Measures for Metal-Organic Networks
Topological index plays an important role in predicting physicochemical properties of a molecular structure. With the help of the topological index, we can associate a single number with a molecular graph. Drugs and other chemical compounds are frequently demonstrated as different polygonal shapes, trees, graphs, etc. In this paper, we will compute irregularity indices for metal-organic networks.
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Graph theory provides the interesting appliance in mathematical chemistry that is used to compute the various kinds of chemical compounds by means of graph theory and predict their various properties [13–19]. One of the most important tools in the chemical graph theory is a topological index, which is useful in predicting the chemical and physical properties of the underlying chemical compound, such as boiling point, strain energy, rigidity, heat of evaporation, and tension [20, 21]. A graph having no loop or multiple edge is known as a simple graph. A molecular graph is a simple graph in which atoms and bounds are represented by the vertex and edge sets, respectively. The degree of the vertex is the number of edges attached with that vertex. These properties of various objects is of primary interest. Winner, in 1947, introduced the concept of the first topological index while finding the boiling point. In 1975, Gutman gave a remarkable identity  about Zagreb indices. Hence, these two indices are among the oldest degree-based descriptors, and their properties are extensively investigated. The mathematical formulae of these indices are
A topological index is known as the irregularity index  if the value of the topological index of the graph is greater than or equal to zero, and the topological index of the graph is equal to zero if and only if the graph is regular. The irregularity indices are given in Table 1. Most of the irregularity indices are from the family of degree-based topological indices and are used in quantitative structure activity relationship modeling.
2. Irregularity Indices for Metal-Organic Networks
In this section, we discuss metal-organic networks by means of a graph. The unit cell of the metal-organic network is given in Figure 1. We give computational results of irregularity indices for two types of metal-organic networks and in the following two sections.
2.1. Irregularity Indices for Metal-Organic Network
This section is about irregularity indices of the metal-organic network . The molecular graph of the metal-organic network for is given in Figure 2. We can observe from Figure 2 that there are four types of vertices present in the molecular graph of , i.e., 2, 3, 4, and 6. The cardinality of vertices 2, 3, 4, and 6 is , 12, , and , respectively. The cardinality of the vertex set of is 48p, i.e., . There are four different types of edges present in the molecular graph of , i.e., , , , and . Their cardinalities are 36, , , and , respectively. The cardinality of the edge set of is , i.e., .
The edge partition of the metal-organic network is given in Table 2.
Theorem 1. Let be the metal-organic network . The irregularity indices are(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)
2.2. Irregularity Indices for Metal-Organic Network
In this section, we will compute irregularity indices for the metal-organic network . The molecular graph of the metal-organic network for is given in Figure 3. It is clear from Figure 3 that there are three types of vertices in the molecular graph of , i.e., 2, 3, and 4. The cardinality of vertices 2, 3, and 4 is , , and , respectively. The cardinality of the vertex set of is 48p, i.e., . There are five different types of edges present in the molecular graph of , i.e., , , , , and . Their cardinalities are , , , , and , respectively. The cardinality of the edge set of is , i.e., .
The edge partition of the metal-organic network is given in Table 3.
Theorem 2. Let be the metal-organic network . The irregularity indices are(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)
3. Graphical Comparison
In this section, we give the graphical comparison of results of the metal-organic networks and . In Figure 4, the red color is fixed for and the green color is fixed for .
Topological indices associate a single number with a chemical structure. In quantitative structure activity relationship, knowledge of topological indices plays an important role. In this article, we computed sixteen irregularity indices for metal-organic networks and . Most of the calculated topological indices depend on degree-based indices. In the field of chemical graph theory, molecular topology, and mathematical chemistry, a degree-based index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. Topological indices are used, for example, in the development of quantitative structure activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. Our results are helpful in drug delivery and computer engineering.
All data required for this research are included within this paper.
Conflicts of Interest
The authors do not have any conflicts of interest.
All authors contributed equally in this paper.
The research was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).
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