Abstract
In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we computed the Hosoya and the Harary polynomials for , and networks. Moreover, we computed serval distance based topological indices, for example, Wiener index, Harary index, and multiplicative version of wiener index.
1. Introduction
In network theory, a network can be taken as graph in which nodes are taken as vertices and links between nodes are taken as edges. Network theory has applications in numerous fields including computer sciences, biology, particle physics, electronics, statistical physics, operational research, finance, sociology, and/or ecology. Uses of network theory incorporate the World Wide Web, strategic systems, internet, quality administrative systems, metabolic systems, epistemological networks, social networks, and so forth.
In the fields of chemical graph theory, chemistry, and molecular topology, a topological index otherwise called a connectivity index is a kind of a molecular descriptor that is computed dependent on the molecular graph of a chemical compound [1, 2]. Topological indices are numerical parameters of a graph which describe its topology and are typically graph invariant. Topological indices are utilized for instance in the improvement of quantitative structure-activity relationships (QSARs) in which the organic action or different properties of compounds are related to their molecular graph [3, 4].
Topological descriptors are obtained from hydrogen-stifled molecular graph, in which the atoms are spoken to by vertices and the bonds by edges. The associations between the atoms can be portrayed by different sorts of topological matrices (e.g., adjacency or distance matric), which can be scientifically controlled in order to infer a solitary number, normally known as graph invariant or topological index [5, 6]. Subsequently, the topological index can be characterized as two-dimensional descriptors that can be effortlessly figured from the molecular graph and do not rely upon the manner in which the graph is delineated or labeled without the energy minimization of the compound structure.
The most simple topological indices do not perceive double bonds and atom types (O, C, Ca, and so forth) and disregard hydrogen atoms and characterized for associated undirected molecular graph only. Polynomials have additionally valuable applications in science; for example, the Hosoya polynomial of a graph is a generating function about distance distributing, introduced by Haruo Hosoya in 1988 [7] and for a connected graph is defined aswhere denotes the distance between vertices and . The Hosoya polynomial has many chemical applications [8], especially, almost all distance based topological indices can be computed from this polynomial [9–11].
The Wiener index of a connected graph is denoted by and defined as the sum of distances between all pairs of vertices in ; i.e., it can be formulated as The Wiener index was firstly introduced by Harold Wiener in 1947 to study the boiling points of paraffin [12]. It plays an important role in the so-called inverse structure-property relationship problems [13]. For more details about this topological polynomial and index, please see the paper series and the references therein [14–22]. Note that the first derivative of the Hosoya polynomial at is equal to the Wiener index:The modified Wiener index of a connected graph is denoted by and defined as the sum of power distances between all pairs of vertices in , where ; i.e., it can be formulated as Hyper-Wiener index is another distance-based graph invariants used for predicting physicochemical properties of organic compounds [23]. The hyper Wiener index was introduced by Randić [24] asModified hyper Wiener index of a connected graph is denoted by and is defined as The Harary polynomial of a connected graph is denoted by and is defined as The generalized Harary index of a connected graph is denoted by and is defined as where The multiplicative Wiener index of a connected graph is denoted by and is defined as In this study, we compute some topological indices of , , , and networks. For details about topological indices and applications of topological indices we refer [25–32].
2. Methodology
A simple graph is a finite nonempty set of objects called vertices together with a (possibly empty) set of unordered pairs of distinct vertices of called edges. To compute Hosoya polynomial of a graph , we need to compute the number of pairs of vertices at distance , where d For this purpose, we use mathematical induction. The general view of Hosoya polynomial is as below:where number of pairs of vertices at distance . By applying fundamental calculus, one can compute Wiener index and hyper Wiener index. Moreover, we use MATLAB for mathematical calculations and verifications (see https://en.wikipedia.org/wiki/MATLAB).
3. Computational Results
In this section, we present our computational results.
3.1. Triangular Oxide Network
A triangulated irregular network (TIN) is a portrayal of a consistent surface comprising completely of triangular aspects, utilized primarily as Discrete Global Grid in primary elevation modeling [33, 34]. Oxide networks play a vital role in the study of silicate networks [35, 36]. If we delete silicon vertices from a silicate network, we get an oxide network . A triangular oxide network is denoted as . An oxide network is depicted in Figure 1, unit cell, Figure 2, for , and Figure 3, for .
Lemma 1. The triangular oxide network has edges and vertices.
Theorem 2. For the triangular oxide network for , where is length of triangular oxide network, we have
Proof. To prove this theorem, we need to compute , where . Obviously, from Lemma 1, we have and The remaining proof is as follows.
It can be observed from Figures 1–3 that there are number of triangles, so Now, one can conclude that Now, for , Now, one can conclude thatNow, for , Now, one can conclude thatFrom the above-mentioned method we can generalize Now, one can conclude that Here, by what have been mentioned above, we have the following computations for the Hosoya polynomial of triangular oxide network .
Theorem 3. For the triangular oxide network , the Harary polynomial is
Proof. From the calculation given in Theorem 2, we have
Theorem 4. For the triangular oxide network , we have (1)(2)(3)(4)
Proof. From the calculation given in Theorem 2, we have the following.
(1) The modified Wiener index is Table 1 contains values of modified Wiener index of for different values of and .
(2) The modified hyper Wiener index is Table 2 contains the values of modified hyper Wiener index of for different values of and .
(3) The generalized Harary index is In Table 3, we give values of generalized Harary index of for different values of and .
(4) The multiplicative Wiener index is
Corollary 5. The Wiener index of triangular oxide network is
Proof. This result can be followed immediately from Theorem 4 by putting in modified Wiener index.
Corollary 6. The hyper Wiener index of triangular oxide network is
Proof. This result can be followed immediately from Theorem 4 by putting in modified hyper wiener index.
Corollary 7. The Harary index of triangular oxide network is
Proof. This result can be followed immediately from Theorem 4 by putting in generalized Harary index.
3.2. Regular Triangular Oxide Network
The regular triangular oxide network is shown in Figure 4, unit cell, Figure 5, for and Figure 6, for
Lemma 8. The regular triangular oxide network has edges and vertices.
Theorem 9. If is the regular triangular oxide network , where is length of triangular oxide network, then
Proof. To prove this theorem, we need to compute where . Obviously from Lemma 8and The remaining proof is as follows.
It can be observed from Figures 4–6 that there are number of triangles: Now, one can conclude thatNow for Now, one can conclude that Now for Now, one can conclude that Now for Now, one can conclude that From the above-mentioned method we can generalize Now, one can conclude thatNow we can calculate for odd distance.
Now for Now, one can conclude that Now for Now, one can conclude that Now for Now, one can conclude that Now for Now, one can conclude that From the above-mentioned method we can generalize Now, one can conclude thatHere, by what have been mentioned above, we have following computations for the Hosoya polynomial of regular triangular oxide network .
Theorem 10. For regular triangular oxide network system , the Harary polynomial is
Proof. From the calculation given in Theorem 9, we have
Theorem 11. For the regular triangular oxide network system , we have + + + + + + + − + + + − + + × ×
Proof. From the calculation given in Theorem 9, we have the following.
(1) The modified Wiener index of regular triangular oxide network is Table 4 contains different values of modified Wiener index of for different values of and .
(2) The modified hyper Wiener index of regular triangular oxide network is Table 5 contains values of modified hyper Wiener index of for different values of and .
(3) The generalized Harary index is Table 6 is about the values of generalized Harary index of for different values of and .
(4) The multiplicative Wiener index is
Corollary 12. The Wiener index of regular triangular oxide network is
Proof. This result immediately follows from Theorem 11 by taking in modified Wiener index.
Corollary 13. The hyper Wiener index of regular triangular oxide network is
Proof. This result immediately follows from Theorem 11 by taking in generalized hyper Wiener index.
Corollary 14. The Harary index of regular triangular oxide network by putting is
Proof. This result immediately follows from Theorem 11 by taking in generalized harary index.
3.3. Triangular Silicate Network
Silicates are the largest, the most complicated, and the most interesting class of minerals so far. tetrahedron is the basic chemical unit of silicates. The silicates sheets are rings of tetrahedrons linked by shared oxygen nodes to other rings in two-dimensional planes producing a sheet like structures. A silicate can be obtained by fusing a metal oxide or a metal carbonate with sand. Essentially every silicate contains tetrahedron. The corner and the center vertices represent oxygen and silicon ions, respectively. These vertices are called oxygen nodes and silicon node, respectively. In tetrahedra, the corner and the center vertices represent oxygen and silicon ions, respectively (Figures 7–9). Unit cell is shown in Figure 7, for in Figure 8 and for in Figure 9.
Lemma 15. The triangular silicate network has edges and vertices.
Theorem 16. If is the triangular silicate network , where is length of triangular silicate network then
Proof. To prove this theorem, we need to compute where . Obviously, from Lemma 15, we have and The remaining proof is as follows.
It can be observed from Figures 7–9 that there are number of triangles: Now, one can conclude that Now, for , Now, one can conclude that Now, for , Now, one can conclude thatFrom the above-mentioned method we can generalize Now, one can conclude thatHere, by what have been mentioned above, we have the following computations for the Hosoya polynomial of triangular oxide network .
Theorem 17. For the triangular silicate network , the Harary polynomial is
Proof. From the calculation given in Theorem 16, we have
Theorem 18. For the triangular silicate network , we have (1)(2)(3)(4)
Proof. From the calculation given in Theorem 16, we have the following.
(1) The modified Wiener index of the triangular silicate network : Table 7 is about different values of modified Wiener index of the triangular silicate network for different values of and .
(2) The modified hyper Wiener index of the triangular silicate network is Table 8 is about different values of modified hyper Wiener index of the triangular silicate network for different values of and
(3) The generalized Harary index the triangular silicate network is Table 9 is about different values of generalized Harary index of the triangular silicate network for different values of and
(4) The multiplicative Wiener index the triangular silicate network is
Corollary 19. The Wiener index of triangular silicate network is
Proof. This result follows immediately Theorem 18 by putting in modified Wiener index.
Corollary 20. The hyper Wiener index of triangular silicate network is
Proof. This result follows immediately Theorem 18 by putting in modified hyper Wiener index.
Corollary 21. The Harary index of triangular Silicate network iswhere is harmonic number.
Proof. This result follows immediately Theorem 18 by putting in generalized Harary index.
3.4. Regular Triangular Silicate Network
The regular triangular silicate network is shown in Figures 10–12, unit cell in Figure 10, for in Figure 11 and for in Figure 12.
Lemma 22. The regular triangular silicate network has edges and vertices.
Theorem 23. If is the regular triangular silicate network , where is length of triangular silicate network then
Proof. To prove this theorem, we need to compute where . Obviously, from Lemma 22, we have and The remaining proof is as follows.
It can be observed from Figures 10–12 that Now, one can conclude thatNow for Now, one can conclude thatNow for Now, one can conclude thatNow for Now, one can conclude thatFrom the above-mentioned method we can generalize Now, one can conclude thatNow we can calculate for odd distance.
For Now, one can conclude thatNow for Now, one can conclude thatNow for Now, one can conclude thatNow for Now, one can conclude thatFrom the above-mentioned method we can generalize Now, one can conclude thatHere, by what have been mentioned above, we have the following computations for the Hosoya polynomial of regular triangular silicate network .
Theorem 24. For regular triangular silicate network system , the Harary polynomial is
Proof. From the information given in Theorem 23, we have
Theorem 25. For the regular triangular silicate network system , we have
Proof. From the information given in Theorem 23, we have the following.
(1) The modified Wiener index of regular triangular silicate network is Table 10 contains the values of modified Wiener index of regular triangular silicate network for different values of and
(2) The modified hyper Wiener index of regular triangular silicate network is Table 11 contains the values of modified hyper Wiener index of regular triangular silicate network for different values of and
(3) The generalized Harary index of regular triangular silicate network is Table 12 contains the values of generalized Harary index of regular triangular silicate network for different values of and
(4) The multiplicative Wiener index of regular triangular silicate network is
Corollary 26. The Wiener index of regular triangular silicate network is
Proof. This result immediately follows from Theorem 25 by putting in modified Wiener index.
Corollary 27. The hyper Wiener index of regular triangular silicate network is
Proof. This result immediately follows from Theorem 25 by putting in modified hyper Wiener index.
Corollary 28. The Harary index of regular triangular silicate network is
Proof. This result immediately follows from Theorem 25 by putting in generalized Harary index.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.