Abstract
A topological index of graph is a numerical quantity which describes its topology. If it is applied to the molecular structure of chemical compounds, it reflects the theoretical properties of the chemical compounds. A number of topological indices have been introduced so far by different researchers. The Wiener index is one of the oldest molecular topological indices defined by Wiener. The Wiener index number reflects the index boiling points of alkane molecules. Quantitative structure activity relationships (QSAR) showed that they also describe other quantities including the parameters of its critical point, density, surface tension, viscosity of its liquid phase, and the van der Waals surface area of the molecule. The Wiener polarity index has been introduced by Wiener and known to be related to the cluster coefficient of networks. In this paper, the Wiener polarity index and Wiener index of certain triangular networks are computed by using graph-theoretic analysis, combinatorial computing, and vertex-dividing technology.
1. Introduction
The Wiener index is originally the first and most studied topological index (see for details in [1]). It was the first molecular topological index that was used in chemistry. Since then, a lot of indices were introduced that relate the topological indices to different physical properties, and some of the recent results can be found in [3–6]. Wiener shows that the Wiener index number is closely correlated with the boiling points of alkane molecules [2]. Later work on quantitative structure activity relationships showed that it is also correlated with other quantities including the parameters of its critical point [7], the density, surface tension, and viscosity of its liquid phase [8], and the van der Waals surface area of the molecule [9].
Mathematically, the Wiener index is sum of all the distances between every vertex of the graph, denoted by , and is
Later on, Wiener introduced another descriptor known as Wiener polarity index that is known to be related to the cluster coefficient of networks. The Wiener polarity index is denoted by and is defined as the number of unordered pairs of vertices that are at distance 3 in . That is,
In organic compounds, say paraffin, the Wiener polarity index is the number of pairs of carbon atoms which are separated by three carbon-carbon bonds. Based on the Wiener index and the Wiener polarity index, the formulawas used to calculate the boiling points of the paraffins, where , , and are constants for a given isomeric group.
By using the Wiener polarity index, Lukovits and Linert demonstrated quantitative structure-property relationships in a series of acyclic and cycle-containing hydrocarbons in [10]. Hosoya in [11] found a physical-chemical interpretation of . Actually, the Wiener polarity index of many kinds of graphs is studied, such as trees [12], unicyclic and bicyclic graphs [13], hexagonal systems, fullerenes, and polyphenylene chains [14], and lattice networks [15]. For more results on the Wiener polarity index, we refer some recent papers [16–19] and the survey paper [20].
2. The Wiener Polarity Index and Wiener Index of Networks Obtained from Triangular Mesh
The graph of the triangular mesh network, denoted by , is obtained inductively by the triangulation of the graph . The procedure to construct this network is as follows:(i)Consider a basic graph which is a cycle of length 3.(ii)Subdivide each edge of , and then join them to form a triangle; the resulting graph is .(iii)Continuing in this way, construct a graph from by subdividing each edge of and then connect them to form triangles.(iv)The graph has vertices on each of its side: The graph of triangular mesh network is shown in Figure 1. The vertices and edges of are defined as follows: .
The count of vertices of the graph is and edges of is .
Furthermore, we partition the vertex set as follows: , where .
Thus, the graph is divided into sets. This will help us in calculating the Wiener and Wiener polarity indices of . The first main result of this chapter is proved in the following.
Theorem 1. , for .
Proof. Now, we find a number of pair of vertices of which are connected through a path of length 3.
However,is the cardinality of the set of vertices in that are at distance 3 from . From the construction of , it is important to note that there is no vertex and such that where . It implies that for and , we have 4 cases to consider.
Case 1. Let , where . If , then ; then, for each , there is only one vertex which is at distance 3 from .
Since ,
Case 2. Let and where . In this case, there are vertices in that are at distance 3 from for each . Since ,
Case 3. Let and where ; in this case, there are vertices in that are at distance 3 from for each . Since ,
Case 4. Let and where . In this case, there are vertices in that are at distance 3 from for each . Since ,Putting equations (6) and (7) and (31) and (9) in (5), we get
In the next result, the Wiener index of the graph is computed.
Theorem 2. .
Proof. Let be the distance between the vertices of and , where .
For ,This can be computed by finding the distance between each vertex from the vertices of . These distances are listed in the following.
For and ,For and ,Thus, we getBy replacing the values of and in equation (11), we getThis after simplification impliesHowever, the Wiener index of is . Therefore,
3. The Wiener Polarity Index and Wiener Index of the Equilateral Triangular Tetra Sheet
This section will start with the definition and properties of the equilateral triangular tetra sheet network. The graph of equilateral triangular tetra sheet network denoted by is obtained from the graph of triangular mesh network by replacing each triangle by the complete graph . This can be done by inserting a vertex in each triangle of the graph and then connecting all the adjacent vertices to form . These new vertices will be denoted by and , where and .
The order and size of the graph are
The graph of equilateral triangular tetra sheet is shown in Figure 2.
In order to compute the Wiener and Wiener polarity indices, we want to find the distance between each pair of vertices of . For this purpose, we define partition of the vertex set as follows: , where , , and .
Furthermore, define , , and .
In the next result, the Wiener polarity index of the graph is computed.
Theorem 3. For , .
Proof. In order to find the Wiener polarity index, we have to compute all those pairs of vertices that are at distance 3 to each other. Since the vertex set of the graph is divided into three parts, we first find the number of such pairs in each possible set. Define as the set of those vertices of that are at distance 3 from the vertices of . For simplicity, . Thus, we haveFor simplicity, we compute the three factors separately:(i)Let and . From the construction of , there does not exist any , s.t. which implies The value of is already calculated in Theorem 1. Furthermore, If , then there are vertices that are at distance 3 from . Thus,(ii)If , there are vertices that are at distance 3 from . From Theorem 1 and equations (22) and (23), we get after simplification(iii)Let and . From the construction of there does not exist any , , s.t. which implies . Thus, we get We compute each of these factors as follows:(iv)If , then there are vertices that are at distance 3 from . Thus, If , then there are vertices of that are at distance 3 from . If , then there are vertices that are at distance 3 from . For every , there are that are at distance 3 from . Substituting each of these values in the second factor, we get after simplification Let and . From the construction of , there does not exist any , s.t. . This implies that , and we getWe compute each of the factors in the following.
If , then there are vertices that are at distance 3 from . This follows thatIf , then there are vertices that are at distance 3 from . This follows thatIf , then there are vertices that are at distance 3 from . This follows thatFor every , there are vertices in that are at distance 3 from .Replace all these values in the third factor, and we get after simplificationBy combining the values of all three factors in equation (20), we found that the Wiener polarity index of the graph is
Theorem 4. .
Proof. Let be the distance between the vertices of from itself and from , , and , where , and . Thus, for any vertex in , we haveThis can further reduce to the following equation:We compute each of the factors in equation (39) separately.
The first factor is computed with the help of following distances: For and , For and , Thus, we get The second factor is computed with the help of following distances: For and , For and , The third factor is computed with the help of following distances: For and , For and , For and , The fourth factor is Putting equations (42), (44), (47), and (48) in (39), we get Let be the distance between the vertices of to itself and from , and , where , is Again, we compute each of the factors separately. The first factor is computed with the help of following distances: For and , For and , The second factor is computed with the help of following distances: For and , For and , The third factor is computed with the help of following distances: For and , For and , The fourth factor isPutting equations (52), (54), (56), and (57) in (50), we getNow, let be the distance between the vertices of to itself and from , where ; then, .
This is equivalent toAgain, we find each factor separately. The first factor is computed with the help of following distances: For and , For and , The second factor is computed with the help of following distances: For and , For , For and , The third factor is computed with the help of following distances: For and , For and , The fourth factor isPutting equations (61), (64), (66), and (67) in (59), we getNow, the Wiener index of the graph is . Hence, by using (49), (58), and (68), we get .
4. The Wiener Polarity Index of the Graph Derived from Hexagonal Networks
The graphs derived from hexagonal networks are finite subgraphs of the triangular grid. In this section, Wiener polarity index of the graph derived from the hexagonal network is computed.
The graph of hexagonal network of dimension is denoted by (Figure 3). The graph contains vertices and edges, where is the number of vertices on one side of the hexagon [5]. There is only one vertex which is at distance from every other corner vertices. This vertex is said to be the center of and is represented by .
Theorem 5. .
Proof. In order to find the vertex index of , firstly, we will divide the graph into two parts by drawing a line passing through the central vertex and parallel to -axis. Now, extend the upper and lower part of the graph to form triangular mesh networks and of dimension , respectively. Define the vertex set of and as follows: and .
It is easy to see that and for . This implies that the Wiener polarity index of can now be written in the following form:From Theorem 1, we know that . This implies thatNow, we calculate the terms , , and , which are equal to the number of vertices of the lower triangle that are at distance 3 from the vertices of the upper triangle. However, for every and , . Therefore,Similarly, for every and , we have .
And, for every and , we haveThis implies thatThe term is the cardinality of set of vertices of that are connected through a path of length 3 from the vertices of . It is easy to see thatThe distance between the set of vertices of the set is equal to and it is easy to see thatNow, by replacing the values of all these factors in equation (69) and simplifying, we get
5. Conclusion
First of all, we will present comparison between two topological indices analytically and graphically.
5.1. Comparison of Wiener and Wiener Polarity Indices of the Triangular Mesh Network
The comparison between the Wiener and Wiener polarity indices of triangular mesh network for different values of is shown in Table 1. The values show that the Wiener index increases rapidly compared to Wiener polarity index as increases. The graphical representation of both indices is also presented. In Figure 4, the black curve denotes the behavior of the Wiener polarity index and red line shows the behavior of the Wiener index.
5.2. Comparison of Wiener and Wiener Polarity Indices of the Graph of Equilateral Triangular Tetra Sheet Networks
The comparison between the Wiener and Wiener polarity indices of equilateral triangular tetra sheet network for different values of is shown in Table 2. The values show that the Wiener index increases rapidly compared to Wiener polarity index as increases. The graphical representation of both indices is also presented. In Figure 5, the black curve denotes the behavior of Wiener polarity index and red line shows the behavior of Wiener index.
In this work, we have derived the Wiener polarity index and Wiener index of certain triangular networks. We have considered triangular grids, equilateral triangular tetra sheets, and hexagonal networks to formulate closed formulas to find the Wiener polarity index and Wiener index. Comparisons of these indices with the help of tables and graphs are also included for two families of the graph. These results will be useful to understand the molecular topology of these important classes of networks.
Data Availability
No data were used for this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is supported by the University Program of Advanced Research (UPAR) and UAEU-AUA grants of United Arab Emirates University (UAEU) via Grant No. G00003271 and Grant No. G00003461.