On Wiener Polarity Index and Wiener Index of Certain Triangular Networks

A topological index of graph G 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. ,eWiener index is one of the oldest molecular topological indices defined by Wiener. ,e 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 derWaals surface area of the molecule.,eWiener polarity index has been introduced by Wiener and known to be related to the cluster coefficient of networks. In this paper, theWiener polarity index Wp(G) andWiener index W(G) of certain triangular networks are computed by using graph-theoretic analysis, combinatorial computing, and vertexdividing technology.


Introduction
e 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][4][5][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 W(G), and is W(G) � p,q∈V (G) d (p, q). (1) Later on, Wiener introduced another descriptor known as Wiener polarity index that is known to be related to the cluster coefficient of networks. e Wiener polarity index is denoted by W p (G) and is defined as the number of unordered pairs of vertices that are at distance 3 in G. at is, W p (G) � (p, q)|d G (p, q) � 3, p, q ∈ V(G) . (2) 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 formula t B � aW(G) + bW p (G) + c (3) was used to calculate the boiling points t B of the paraffins, where a, b, and c 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 W p (G). 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][17][18][19] and the survey paper [20].

The Wiener Polarity Index and Wiener
Index of Networks Obtained from Triangular Mesh e graph of the triangular mesh network, denoted by T n , is obtained inductively by the triangulation of the graph T n−1 . e procedure to construct this network is as follows: (i) Consider a basic graph T 3 which is a cycle C 3 of length 3. (ii) Subdivide each edge of T 3 , and then join them to form a triangle; the resulting graph is T 4 . (iii) Continuing in this way, construct a graph T n from T n−1 by subdividing each edge of T n−1 and then connect them to form triangles. (iv) e graph T n has n vertices on each of its side: e graph of triangular mesh network T 5 is shown in Figure 1.
e vertices and edges of T n are defined as follows: e count of vertices of the graph T n is n(n + 1)/2 and edges of T n is 3n(n − 1)/2. Furthermore, we partition the vertex set V(T n ) as follows: us, the graph is divided into n sets. is will help us in calculating the Wiener and Wiener polarity indices of T n . e first main result of this chapter is proved in the following.
Proof. Now, we find a number of pair (p, q) of vertices of T n which are connected through a path of length 3. However, is the cardinality of the set of vertices in V m that are at distance 3 from V l . From the construction of T n , it is important to note that there is no vertex p ∈ V l and q ∈ V m such that d(p, q) � 3 where l, m ∈ 1, 2, 3 { }. It implies that for x ∈ V l and y ∈ V m , we have 4 cases to consider.
If d(x, y) < 3, then |l − m| ≤ 3; then, for each l, there is only one vertex x l,m+3 which is at distance 3 from x l,m . Since l ≤ m ≤ n − 3, Case 2. Let u ∈ V l and v ∈ V l−1 where 4 ≤ l ≤ n. In this case, there are 2l − 6 vertices in V l−1 that are at distance 3 from V l for each i. Since i ≤ m ≤ n − 3, x 1,1   Journal of Chemistry Case 3. Let u ∈ V l and v ∈ V l−2 where 4 ≤ l ≤ n; in this case, there are 2l − 6 vertices in V l−2 that are at distance 3 from V l for each i. Since i ≤ m ≤ n − 3, Case 4. Let u ∈ V l and v ∈ V l−3 where 4 ≤ l ≤ n. In this case, there are 4l + 6 vertices in V l−3 that are at distance 3 from V l for each i. Since i ≤ m ≤ n − 3, Putting equations (6) and (7) and (31) and (9) in (5), we get In the next result, the Wiener index of the graph T n is computed.
Proof. Let W(V n , T n−1 ) be the distance between the vertices of V n and T n−1 , where V n � x n,1 , x n,2 , x n,3 , . . . , x n,n .
For x lm ∈ V(T n−1 ), is can be computed by finding the distance between each vertex x n,θ from the vertices of T n−1 . ese distances are listed in the following.

The Wiener Polarity Index and Wiener Index of the Equilateral Triangular Tetra Sheet
is section will start with the definition and properties of the equilateral triangular tetra sheet network. e graph of equilateral triangular tetra sheet network denoted by ET n is obtained from the graph of triangular mesh network by replacing each triangle by the complete graph K 4 . is can be done by inserting a vertex in each triangle of the graph T n and then connecting all the adjacent vertices to form K 4 .
ese new vertices will be denoted by u i and w j , where 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n − 2.
e order and size of the graph ET(n) are e 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 ET n . For this purpose, we define partition of the vertex set as follows: In the next result, the Wiener polarity index of the graph ET n is computed.
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 ET n is divided into three parts, we first find the number of such pairs in each possible set. Define W p (A, B) as the set of those vertices of A that are at distance 3 from the vertices of B. For simplicity, W p (A, A) � W p (A). us, we have For simplicity, we compute the three factors separately: From eorem 1 and equations (22) and (23), we get after simplification Journal of Chemistry 5 We compute each of these factors as follows: If x ∈ W l−4 , then there are 2l − 6 vertices that are at distance 3 from U l−1 . (28) Substituting each of these values in the second factor, we get after simplification Let w ∈ W l−2 and x ∈ W l−2 ′ . From the construction of ET n , there does not exist any Journal of Chemistry We compute each of the factors in the following.
then there are 2l − 4 vertices that are at distance 3 from W l−2 . is follows that If x ∈ U l−3 ∪ V l−3 , then there are 4l − 6 vertices that are at distance 3 from W l−2 . is follows that If x ∈ W l−4 , then there are 3l − 6 vertices that are at distance 3 from W l−2 . is follows that For every w ∈ W l−2 , there are l − 2 vertices x in w ′ ∈ W l−2 that are at distance 3 from w.
Replace all these values in the third factor, and we get after simplification l�n−2 l�4 W p W l−2 + W p W l−2 , W l−2 ′ � 10n 2 − 66n + 108.

(36)
By combining the values of all three factors in equation (20), we found that the Wiener polarity index of the graph ET n is Proof. Let W(V n , ET n ′ ) be the distance between the vertices of V n from itself and from U n−1 , W n−2 , and ET n−1 , where V n � x n,1 , x n,2 , x n,3 , . . . , x n,n , U n−1 � u n,1 , u n,2 , u n,3 , . . . , u n,n−1 }, and W n−1 � w n,1 , w n,2 , w n,3 , . . . , w n,n−2 .
us, for any vertex in V n , we have We compute each of the factors in equation (39) separately.
e first factor is computed with the help of following distances: e second factor is computed with the help of following distances: e third factor is computed with the help of following distances: For θ � n and 1 ≤ m ≤ θ − 2, d x n,θ , w l,m � n − m for m ≤ l ≤ n − 2.
For 1 ≤ θ ≤ n − 2 and θ ≤ m ≤ n − 2, (50) Again, we compute each of the factors separately. e first factor is computed with the help of following distances: For 2 ≤ θ ≤ n − 2 and 1 ≤ m ≤ θ − 1, (52) e second factor is computed with the help of following distances: (54) e third factor is computed with the help of following distances: e fourth factor is Putting equations (52), (54), (56), and (57) in (50), we get W U n−1 , ET n″ � 1 12 Now, let W(W n−2 , ET n′″ ) be the distance between the vertices of W n−2 to itself and from ET n−1 , where W n−2 � w n−2,1 , w n−2,2 , w n−2,3 , . . . , w n−2,n−2 ; then, W(W n−2 , Again, we find each factor separately. e first factor is computed with the help of following distances: For 1 ≤ θ ≤ n − 2 and 1 ≤ m ≤ θ, e second factor is computed with the help of following distances:  (64) e third factor is computed with the help of following distances:

The Wiener Polarity Index of the Graph
Derived from Hexagonal Networks e 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. e graph of hexagonal network of dimension n is denoted by HX n (Figure 3). e graph contains 3n 2 − 3n + 1 vertices and 9n 2 − 15n + 6 edges, where n is the number of vertices on one side of the hexagon [5]. ere is only one vertex v which is at distance n − 1 from every other corner vertices. is vertex is said to be the center of HX n and is represented by O.
Proof. In order to find the vertex PI index of HX n , firstly, we will divide the graph into two parts by drawing a line passing through the central vertex and parallel to x-axis. Now, extend the upper and lower part of the graph HX n to form triangular mesh networks T 2n−1 and T 2n−1 ′ of dimension 2n − 1, respectively. Define the vertex set of T 2n−1 and T 2n−1 ′ as follows: It is easy to see that V(T 2n−1 ) ∩ V(T 2n−1 ′ ) � V 2n−1 and d(x lm , v lm ′ ) > 3 for l < 2n − 3. is implies that the Wiener polarity index of HX n can now be written in the following form: From eorem 1, we know that W p (T n ) � 9(n 2 − 5n + 6)/2. is implies that Now, we calculate the terms W p (V 2n−2 , V 2n−2 ′ ), W p (V 2n−2 , V 2n−3 ′ ), and W p (V 2n−3 , V 2n−2 ′ ), 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, Similarly, for every v ∈ V 2n−2 and v ′ ∈ V 2n−3 ′ , we have And, for every v ∈ V 2n−3 and v ′ ∈ V 2n−2 ′ , we have is implies that HX 5 T 2n-1 x 11 x 2n-1,1 x 2n-1,2n-1 T n-1 Figure 3: Hexagonal network HX n and its extension. e term W p (HX n , T n−1 ) is the cardinality of set of vertices of T n−1 that are connected through a path of length 3 from the vertices of HX n . It is easy to see that W p HX n , T n−1 � (74) e distance between the set of vertices of the set V 2n−1 is equal to W p (V 2n−1 ) and it is easy to see that Now, by replacing the values of all these factors in equation (69) and simplifying, we get W p HX n � 2 W p T 2n−1 − Wp HX n , T n−1 − W p T n−1 + W p V 2n−2 , V 2n−2 ′ + W p V 2n−2 , V 2n−3 ′ + W p V 2n−3 , V 2n−2 ′ − W p V 2n−1 � 27n 2 − 81n + 48. (76)

Conclusion
First of all, we will present comparison between two topological indices analytically and graphically.

Comparison of Wiener and Wiener Polarity Indices of the Triangular Mesh Network.
e comparison between the Wiener and Wiener polarity indices of triangular mesh network T n for different values of n is shown in Table 1. e values show that the Wiener index increases rapidly compared to Wiener polarity index as n increases. e 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.

Comparison of Wiener and Wiener Polarity Indices of the Graph of Equilateral Triangular Tetra Sheet Networks.
e comparison between the Wiener and Wiener polarity indices of equilateral triangular tetra sheet network ET n for different values of n is shown in Table 2. e values show that the Wiener index increases rapidly compared to Wiener polarity index as n increases. e 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. ese 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
e authors declare that they have no conflicts of interest.