Abstract
Structure-based topological descriptors of chemical networks enable us the prediction of physico-chemical properties and the bioactivities of compounds through QSAR/QSPR methods. Topological indices are the numerical values to represent a graph which characterises the graph. One of the latest distance-based topological index is the Mostar index. In this paper, we study the Mostar index, Szeged index, PI index, index, and index, for chain oxide network , chain silicate network , ortho chain , and para chain , for the first time. Moreover, analytically closed formulae for these structures are determined.
1. Introduction and Preliminary Results
All the graphs in this paper are considered to be finite, undirected, and loopless. Graph is the set made up of vertices (also called the nodes) which are connected with the edges (also called links). It consists on two sets and , where is called the vertex set and is called the edge set. In order to understand the properties and information contained in the connectivity pattern of graphs, there are many numbers of numerical quantities, known as structure invariants, topological indices, or topological descriptors, which have been derived and studied over the past few decades. The topological indices have vast number of applications in the chemical graph theory which is the special branch of mathematical chemistry.Graph theory has a wide range of applications in engineering due to its diagrammatic nature. It is used in computer science to study the algorithms and flow of information. In engineering, it is used to model the graphics and designs of different networks by converting them in the form of graph.
The topological indices are very much used for characterizing the chemical graphs on the basis of their numerical values. They establish the relationship between the structure and properties of molecule. Topological indices are widely used in QSAR and QSPR research studies [1]. Till now, many topological indices have been derived. For any two graphs and which are isomorphic to each other, then [2]. Due to the success of simple topological indices, such as Wiener Index [3], Zagreb index [4], and Szeged index [5], motivated others, hundreds of topological indices are introduced. Wiener index is one of the first index which was introduced by Harold Wiener in 1947 [6], when he was working on the boiling point of paraffins. The Wiener index [7] of a graph is defined as the sum of all the distances between pairs of vertices of :where denotes the shortest-path distance in .
The Szeged index is defined aswhere denotes the number of vertices of closer to than to and is defined as the number of vertices of closer to than to . This was first studied by Gutman. Later, it is known as the Szeged index [8].
The PI index [9], of a graph , is defined as
The Graovac–Ghorbani index is defined asand this index is introduced by Graovac and Ghorbani [10], and Furtula [11] used the name Graovac–Ghorbani index.
The normalized index is index, first studied by Dimitrov et al. [12], and is defined as
A chemical graph is a simple graph in which atoms correspond to the vertices and edge denotes the bond between two atoms. A topological index, specially, the Mostar index is one of the latest topological index, derived in 2018 [13]. Previously, Arockiaraj [14] found the Mostar indices of carbon nanostructures, and Hayata and Zhou [15] calculated the large Mostar index on cacti. The Mostar index for a graph is defined as the sum of all the absolutes values of the difference between and , where and are the adjacent vertices of an edge:
2. Main Results
The main goal of this article is to compute the Mostar index of ortho chain and para chain using the edge cut method; also, we find the Mostar index, Szeged index, PI index, index, and index of oxide chains, chain silicates, ortho chain, and para chain by using the technique of edge partition. The notations used in this paper are standard and taken from the book of west [16]. For the concepts and terms not defined here, we refer the reader to concern with the book of Harary [17] and also concern with [18–25].
2.1. Results for the Chain Oxide Network
In this section, we discuss and compute the exact results for Szeged, PI, , , and Mostar index. If we remove the silicon atom from the silicate network, then the resulting network is an oxide network [26], which consists of three oxygen atoms. Oxide network has the triangular structure. If an oxide network shares its oxygen with other oxide network linearly, then the oxide chain is formed, as shown in Figure 1.
Theorem 1. Let be the oxide network of order, then its Szeged index is .
Proof. Let , where ; also, is an integer.By using Table 1, we havewhich is required.
Theorem 2. Let be the oxide network of order; then, its PI index is .
Proof. Let , where ; also, is an integer:By using Table 2, we havewhich is required.
Theorem 3. Let be the oxide network of order; then, its index is .
Proof. Let , where ; also, is an integer:By using Table 3, we havewhich is required.
Theorem 4. Let be the oxide network of order; then, its index is .
Proof. Let , where ; also, is an integer:By using Table 4, we havewhich is required.
Theorem 5. Let be the oxide network of even order; then, its Mostar index is .
Proof. Let , where ; also, is even:By using Table 5, we havewhich is required.
Theorem 6. Let be the oxide network of odd order; then, its Mostar index is .
Proof. Let , where ; also, is odd:By using Table 6, we havewhich is required.
2.2. Results for the Chain Silicate Network
In this section, we discuss and compute the exact results for Szeged, PI, , , and Mostar index. Silicates are the compounds which consist of silicon and oxygen, having the tetrahedron structure with bond angle of . is found in almost all of the silicates. A single tetrahedron has a shape like a pyramid with triangular base. It has four oxygen atoms at its corners, and silicon atom is bounded equally with oxygen atoms with bond length of . A single tetrahedron is shown in Figure 2(a). If a single tetrahedron shares its oxygen with other tetrahedrons; then, a linear silicate chain [27] is formed, as shown in Figure 2(b).
(a)
(b)
Theorem 7. Let be the chain silicate network of order; then, its Szeged index is .
Proof. Let , where ; also, is an integer:By using Table 7, we havewhich is required.
Theorem 8. Let be the chain silicate network of order; then, its PI index is .
Proof. Let , where ; also, is an integer:By using Table 8, we havewhich is required.
Theorem 9. Let be the chain silicate network of order then its index is .
Proof. Let , where ; also, is an integer:By using Table 9, we havewhich is required.
Theorem 10. Let be the chain silicate network of order; then, its index is .
Proof. Let , where ; also, is an integer:By using Table 10, we havewhich is required.
Theorem 11. Let be the chain silicate network of even order; then, its Mostar index is .
Proof. Let , where ; also, is even.By using Table 11, we havewhich is required.
Theorem 12. Let be the chain silicate network of odd order; then, its Mostar index is .
Proof. Let , where ; also, is odd:By using Table 12, we havewhich is required.
2.3. Results for the Ortho Chain
In this section, we discuss and compute the exact results for Szeged, PI, , , and Mostar index. The single molecule of para and ortho chain has the same structure. Basically, it is a cycle graph having 4 sides denoted as and represented as a four-sided regular polygon. The ortho chain has a zig-zag structure where each corner of is attached linearly, as shown in Figure 3. The para chain has a structure in which each is attached at corner to corner with other but not linearly, as shown in Figure 4 [28].
Theorem 13. Let be the ortho chain of order; then, its Szeged index is .
Proof. Let , where ; also, is an integer.By using Table 13, we havewhich is required.
Theorem 14. Let be the ortho chain of order; then, its PI index is .
Proof. Let , where ; also, is an integer:By using Table 14, we havewhich is required.
Theorem 15. Let be the ortho chain of order; then, its index is .
Proof. Let , where ; also, is an integer:By using Table 15, we havewhich is required.
Theorem 16. Let be the ortho chain of order; then, its index is .
Proof. Let , where ; also, is an integer:By using Table 16, we havewhich is required.
Theorem 17. Let be the ortho chain of even order; then, its Mostar index is .
Proof. Let , where ; also, is even:By using Table 17, we havewhich is required.
Theorem 18. Let be the ortho chain of odd order; then, its Mostar index is .
Proof. Let , where ; also, is odd:By using Table 18, we havewhich is required.
2.4. Results for the Para Chain
In this section, we discuss and compute the exact results for Szeged, PI, , , and Mostar index.
Theorem 19. Let be the para chain of order; then, its Szeged index is .
Proof. Let , where ; also, is an integer.By using Table 19, we havewhich is required.
Theorem 20. Let be the para chain of order; then, its PI index is .
Proof. Let , where ; also, is an integer:By using Table 20, we havewhich is required.
Theorem 21. Let be the para chain of order; then, its index is .
Proof. Let , where ; also, is an integer:By using Table 21, we havewhich is required.
Theorem 22. Let be the para chain of order; then, its index is .
Proof. Let , where also is an integer:By using Table 22, we havewhich is required.
Theorem 23. Let be the para chain of even order; then, its Mostar index is .
Proof. Let , where ; also, is even:By using Table 23, we havewhich is required.
Theorem 24. Let be the para chain of odd order; then, its Mostar index is .
Proof. Let , where ; also, is odd:By using Table 24, we havewhich is required. For the comparison of Szeged, PI, , and index of , we computed the indices for different values of . By increasing the values of , we can clearly check from Table 25 that the order of Szeged and PI index is increasing while that of and is decreasing. For the comparison of Szeged, PI, , and index of , we computed the indices for different values of . By increasing the values of , we can clearly check from Table 26 that the order of Szeged and PI index is increasing while that of and is decreasing. For the comparison of Szeged, PI, , and index of , we computed the indices for different values of . By increasing the values of , we can clearly check from Table 27 that the order of Szeged and PI index is increasing while that of and is decreasing. For the comparison of Szeged, PI, , and index of , we computed the indices for different values of . By increasing the values of , we can clearly check from Table 28 that the order of Szeged and PI index is increasing while that of and is decreasing.
3. Conclusion
In this article, we have figured out several bond-additive TIs such as Szeged, PI, , , and Mostar index. We calculated the closed formulae for abovementioned TIs of chain silicate, oxide network, para, and ortho chain. The above outcomes contribute in the field of natural sciences and pharmaceutical science. Our exploration kept on determining new consequences of these graphs.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.