Research Article  Open Access
Lian Chen, Abid Mehboob, Haseeb Ahmad, Waqas Nazeer, Muhammad Hussain, M. Reza Farahani, "Hosoya and Harary Polynomials of and ", Discrete Dynamics in Nature and Society, vol. 2019, Article ID 8696982, 18 pages, 2019. https://doi.org/10.1155/2019/8696982
Hosoya and Harary Polynomials of and
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 structureactivity 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 hydrogenstifled 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 twodimensional 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 socalled inverse structureproperty 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 HyperWiener index is another distancebased 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 abovementioned 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 abovementioned 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 abovementioned 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

