# On Wiener Polarity Index and Wiener Index of Certain Triangular Networks

Authors: Mr. Adnan; Syed Ahtsham Ul Haq Bokhary; Muhammad Imran
Journal: Journal of Chemistry
 (2021)

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## Abstract

A topological index of graphG 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 WpG and Wiener index WG of certain triangular networks are computed by using graph-theoretic analysis, combinatorial computing, and vertex-dividing technology.


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## Body

## 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 byWG, and is(1)WG=∑p,q∈VGdp,q.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 byWpG and is defined as the number of unordered pairs of vertices that are at distance 3 in G. That is,(2)WpG=p,q|dGp,q=3,p,q∈VG.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(3)tB=aWG+bWpG+cwas used to calculate the boiling points tB 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 WpG. 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 byTn, is obtained inductively by the triangulation of the graph Tn−1. The procedure to construct this network is as follows:(i)
Consider a basic graphT3 which is a cycle C3 of length 3.(ii)
Subdivide each edge ofT3, and then join them to form a triangle; the resulting graph is T4.(iii)
Continuing in this way, construct a graphTn from Tn−1 by subdividing each edge of Tn−1 and then connect them to form triangles.(iv)
The graphTn has n vertices on each of its side:The graph of triangular mesh networkT5 is shown in Figure 1.The vertices and edges ofTn are defined as follows:VTn=xl,m:1≤l≤n,1≤m≤l.ETn=xl,mxl,m+1:2≤l≤n,1≤m≤l−1∪xl,m,xl+1,m:1≤l≤n−1,1≤m≤l∪xl,m,xl+1,m+1:1≤l≤n−1,2≤m≤l+1.Figure 1
Triangular mesh.The count of vertices of the graphTn is nn+1/2 and edges of Tn is 3nn−1/2.Furthermore, we partition the vertex setVTn as follows: VTn=∪l=1nVl, where Vl=xl,m:1≤l≤n1≤m≤l.Thus, the graph is divided inton sets. This will help us in calculating the Wiener and Wiener polarity indices of Tn. The first main result of this chapter is proved in the following.Theorem 1.
WpTn=9n−2n−3/2, for n≥3.Proof.
Now, we find a number of pairp,q of vertices of Tn which are connected through a path of length 3.
However,(4)WpVl=p,q|dp,q=3;p,q∈VlWpVl,Vm=p,q|dp,q=3;p∈Vl,q∈Vmis the cardinality of the set of vertices in Vm that are at distance 3 from Vl. From the construction of Tn, it is important to note that there is no vertex p∈Vl and q∈Vm such that dp,q=3 where l,m∈1,2,3. It implies that for x∈Vl and y∈Vm, we have 4 cases to consider.(5)WpTn=∑l=4l=nWpVl+∑l=4l=nWpVl,Vl−1+∑l=4l=nWpVl,Vl−2+∑l=4l=nWpVl,Vl−2.Case 1.
Letxlm∈Vl, where 4≤l≤n. If dx,y<3, then l−m≤3; then, for each l, there is only one vertex xl,m+3 which is at distance 3 from xl,m.
Sincel≤m≤n−3,(6)WpVl=p,q|dp,q=3;p,q∈Vl=l−3∑l=4l=nWpVl=∑l=4l=nl−3=n−3n−22.Case 2.
Letu∈Vl and v∈Vl−1 where 4≤l≤n. In this case, there are 2l−6 vertices in Vl−1 that are at distance 3 from Vl for each i. Since i≤m≤n−3,(7)WpVl,Vl−1=2l−6∑l=4l=nWpVl,Vl−1=∑l=4l=n2l−6=n−2n−3.Case 3.
Letu∈Vl and v∈Vl−2 where 4≤l≤n; in this case, there are 2l−6 vertices in Vl−2 that are at distance 3 from Vl for each i. Since i≤m≤n−3,(8)WpVl,Vl−2=2l−6∑l=4l=nWpVl,Vl−2=∑l=4l=n2l−6=n−2n−3.Case 4.
Letu∈Vl and v∈Vl−3 where 4≤l≤n. In this case, there are 4l+6 vertices in Vl−3 that are at distance 3 from Vl for each i. Since i≤m≤n−3,(9)WpVl,Vl−3=4l+6∑l=4l=nWpVl,Vl−2=∑l=4l=n4l+6=2n−2n−3.
Putting equations (6) and (7) and (31) and (9) in (5), we get(10)WpTn=n−2n−32+n−2n−3+n−2n−3+2n−2n−3=9n−2n−32.In the next result, the Wiener index of the graphTn is computed.Theorem 2.
WTn=WTn−1+n4−n2/4.Proof.
LetWVn,Tn−1 be the distance between the vertices of Vn and Tn−1, where Vn=xn,1,xn,2,xn,3,…,xn,n.
Forxlm∈VTn−1,(11)WVn,Tn−1=∑θ=1θ=nWxn,θ,Tn−1+WVn.
This can be computed by finding the distance between each vertexxn,θ from the vertices of Tn−1. These distances are listed in the following.
For2≤θ≤n and 1≤m≤θ−1,(12)dxn,θ,xl,m=n−1,form≤l≤n+m−θ−1,θ−m,forn+m−θ≤l≤n−1.
For1≤θ≤n and θ≤m≤n,(13)dxn,θ,xl,m=n+m−l−θform≤l≤n−1.
Thus, we get(14)∑θ=1θ=nWxn,θ,Tn−1=∑θ=1θ=n∑m=1m=θ−1∑l=ml=n−1dxn,θ,xlm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n−1dxn,θ,xlm+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1dxn,θ,xlm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n+m−θ−1dxn,θ,xlm+∑θ=2θ=n∑m=1m=θ−1∑l=n+m−θl=n−1dxn,θ,xlm+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1dxn,θ,xlm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n+m−θ−1n−l+∑θ=2θ=n∑m=1m=θ−1∑l=n+m−θl=n−1θ−m+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1n+m−l−θ=112n4−2n3−n2+2n+112n4−n2+112n4−n2,(15)WVn=∑θ=1θ=nWxn,θ=∑l=1l=n−1∑m=lm=n−lm=16n3−n.
By replacing the values of∑θ=1θ=nWxn,θ,Tn−1 and WVn in equation (11), we get(16)WVn,Tn−1=112n4−2n3+2n−n2+112n4−n2+112n4−n2+16n3−n.
This after simplification implies(17)WVn,Tn−1=14n4−n2.
However, the Wiener index ofTn is WTn=WTn−1+WVn,Tn−1. Therefore,(18)WTn=WTn−1+14n4−n2.


## 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 byETn is obtained from the graph of triangular mesh network by replacing each triangle by the complete graph K4. This can be done by inserting a vertex in each triangle of the graph Tn and then connecting all the adjacent vertices to form K4. These new vertices will be denoted by ui and wj, where 1≤i≤n−1 and 1≤j≤n−2.The order and size of the graphETn are(19)VETn=3n2−3n+22,EETn=9n2−15n+62.The graph of equilateral triangular tetra sheet is shown in Figure2.Figure 2
The graph of equilateral triangular tetra sheet.In order to compute the Wiener and Wiener polarity indices, we want to find the distance between each pair of vertices ofETn. For this purpose, we define partition of the vertex set as follows: VETn=∪l=1l=nVl∪∪l=1l=n−1Ul∪∪l=1l=n−2Wl, where Vl=xlm|1≤m≤l≤n, Ul=ulm|1≤m≤l≤n−1, and Wl=wlm|1≤m≤l≤n−2.Furthermore, defineVl′=VETl∖Vl, Ul′=VETl∖Vl∪Ul, and Wl′=VETl∖Vl+2∪Ul+1∪Wl.In the next result, the Wiener polarity index of the graphETn is computed.Theorem 3.
Forn≥3, WpETn=63n2−357n+504/2.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 graphETn is divided into three parts, we first find the number of such pairs in each possible set. Define WpA,B as the set of those vertices of A that are at distance 3 from the vertices of B. For simplicity, WpA,A=WpA. Thus, we have(20)WpVl=v,v′|dv,v′=3;v,v′∈VlWpUl=u,u′|du,u′=3;u,u′∈UlWpWl=w,w′|dw,w′=3;w,w′∈WlWpVl,Vl′=v,x|dv,x=3;v∈Vl,x∈Vl′;4≤l≤nWpUl,Ul′=u,x|du,x=3;u∈Ul,x∈Ul′;3≤l≤n−1WpWl,Wl′=w,x|dw,x=3;w∈Wl,x∈Wl′;2≤l≤n−2WpETn=∑l=4l=nWpVl+WpVl,Vl′+∑l=3l=n−1WpUl+WpUl,Ul′+∑l=2l=n−2WpWl+WpWl,Wl′.
For simplicity, we compute the three factors separately:(i)
Letv∈Vl and x∈Vl′. From the construction of ETn, there does not exist any x∈Vl−θ′, 4≤θ≤n−1 s.t. dv,x=3 which implies x∈Vl−1∪Vl−2∪Vl−3∪Ul−1∪Ul−2∪Ul−3∪Wl−2∪Wl−3∪Wl−4The value of∑l=1l=n−1WpVl is already calculated in Theorem 1. Furthermore,(21)WpVl,Vl′=WpVl,Ul−1+WpVl,Wl−2+WpVl,Vl−1+WpVl,Ul−2+WpVl,Wl−3+WpVl,Vl−3+WpVl,Ul−3+WpVl,Wl−4.Ifx∈Ul−1∪Wl−2∪Ul−2∪Wl−3, then there are 2l−6 vertices that are at distance 3 from Vl. Thus,(22)WpVl,Ul−1=WpVl,Wl−2=WpVl,Ul−2=WpVl,Wl−3=2l−6∑l=4l=nWpVl,Ul−1=∑l=4l=nWpVl,Wl−2=∑l=4l=nWpVl,Ul−2=∑l=4l=n=WpVl,Wl−3∑l=4l=n2l−6=n−2n−3.(ii)
Ifx∈Ul−3∪Wl−4, there are 3l−12 vertices that are at distance 3 from Vl.(23)WpVl,Ul−3=WpVl,Wl−4=3l−12∑l=4l=nWpVl,Ul−3=∑l=4l=nWpVl,Wl−4=∑l=4l=n3l−12=3n2−21n+362.From Theorem1 and equations (22) and (23), we get after simplification(24)∑l=4l=nWpVl+WpVl,Vl′=3n−34n−9.(iii)
Letu∈Ul−1 and x∈Ul−1′. From the construction of ETn there does not exist any x∈Ul−θ′, 4≤θ≤n−1, s.t. du,x=3 which implies x∈Vi∪Vl−1∪Vl−2∪Ul−1∪Ul−2∪Ul−3∪Wl−2∪Wl−3∪Wl−4. Thus, we get(25)WpUl−1,Ul−1′=WpUl−1,Wl−2+WpUl−1,Vl−1+WpUl−1,Ul−2+WpUl−1,Wl−3+WpUl−1,Vl−2+WpUl−1,Ul−3+WpUl−1,Wl−4+WpUl−1,Vl−3.We compute each of these factors as follows:(iv)
Ifx∈Wl−2∪Vl−1∪Ul−2∪Wl−3∪Vl−2, then there are 2l−4 vertices x that are at distance 3 from Ul−1. Thus,(26)WpUl−1,Wl−2=WpUl−1,Vl−1=WpUl−1,Ul−2=WpUl−1,Wl−3=WpUl−1,Vl−2=2l−4,∑l=3l=n−1WpUl,Wl−2=∑l=3l=n−1WpUl−1,Vl−1=∑l=3l=n−1WpUl−1,Ul−2=∑l=3l=n−1WpUl−1,Wl−3=∑l=3l=n−1WpUl−1,Vl−2=∑l=3l=n−12l−4=n−2n−3.Ifx∈Ul−3∪Vl−3, then there are 3l−6 vertices of x that are at distance 3 from Ul−1.(27)WpUl−1,Ul−3=WpUl−1,Vl−3=2l−6∑l=3l=n−1WpUl−1,Ul−3=∑l=3l=n−1WpUl−1,Vl−3=∑l=3l=n−12l−6=123n2−15n+18.Ifx∈Wl−4, then there are 2l−6 vertices that are at distance 3 from Ul−1.(28)WpUl−1,Wl−4=2l−6∑l=4l=n−1WpUl−1,Wl−4=∑l=4l=n−12l−6=n2−7n+12.For everyu∈Ul−1, there are l−2u′∈Ul−1 that are at distance 3 from u.(29)WpUl−1=l−2∑l=3l=n−1WpUl−1=∑l=3l=n−1l−2=n2−5n+62.Substituting each of these values in the second factor, we get after simplification(30)∑l=4l=n−1WpUl−1+WpUl−1,Ul−1′=192n−21n−3.Letw∈Wl−2 and x∈Wl−2′. From the construction of ETn, there does not exist any x∈Ul−θ′, 4≤θ≤n−1 s.t. du,x=3. This implies that x∈Vl−1∪Vl−2∪Ul−2∪Ul−3∪Wl−3∪Wl−4, and we get(31)WpWl−2,Wl−2′=WpWl−2,Vl−1+WpWl−2,Ul−2+WpWl−2,Wl−3+WpWl−2,Vl−2+WpWl−2,Ul−3+WpWl−2,Wl−4+WpWl−2,Vl−3.
We compute each of the factors in the following.
Ifx∈Vl−1∩Ul−2∪Wl−3∪Vl−2, then there are 2l−4 vertices that are at distance 3 from Wl−2. This follows that(32)WpWl−2,Vl−1=WpWl−2,Ul−2=WpWl−2,Wl−3=WpWl−2,Vl−2=2l−4,∑l=3l=n−2WpWl−2,Vl−1=∑l=3l=n−2WpWl−2,Ul−2=∑l=3l=n−2WpWl−2,Wl−3=∑l=3l=n−2WpWl−2,Vl−2=∑l=3l=n−22l−4=n2−7n+12.
Ifx∈Ul−3∪Vl−3, then there are 4l−6 vertices that are at distance 3 from Wl−2. This follows that(33)WpWl−2,Ul−3=WpWl−2,Vl−3=4l−6,∑l=3l=n−2WpWl−2,Ul−3=∑l=3l=n−2WpWl−2,Vl−3=∑l=3l=n−24l−6=2n2−12n+18.
Ifx∈Wl−4, then there are 3l−6 vertices that are at distance 3 from Wl−2. This follows that(34)WpWl−2,Wl−4=3l−6∑l=3l=n−2WpWl−2,Wl−4=∑l=3l=n−23l−6=3n2−21n+362.
For everyw∈Wl−2, there are l−2 vertices x in w′∈Wl−2 that are at distance 3 from w.(35)WpWl−1=l−2,∑l=3l=n−2WpWl−1=∑l=3l=n−2l−2=n2−7n+122.
Replace all these values in the third factor, and we get after simplification(36)∑l=4l=n−2WpWl−2+WpWl−2,Wl−2′=10n2−66n+108.
By combining the values of all three factors in equation (20), we found that the Wiener polarity index of the graph ETn is(37)WpETn=63n2−357n+5042.Theorem 4.
WETn=WETn−1+1/1228n4−86n3+134n2−172n+120.Proof.
LetWVn,ETn′ be the distance between the vertices of Vn from itself and from Un−1, Wn−2, and ETn−1, where Vn=xn,1,xn,2,xn,3,…,xn,n,Un−1=un,1,un,2,un,3,…,un,n−1, and Wn−1=wn,1,wn,2,wn,3,…,wn,n−2. Thus, for any vertex in Vn, we have(38)WVn,ETn′=∑θ=1θ=nWxn,θ,ETn−1+∑θ=1θ=nWxn,θ,Un−1+∑θ=1θ=nWxn,θ,Wn−2+WVn.
This can further reduce to the following equation:(39)WVn,ETn′=∑θ=1θ=nWxn,θ,∪l=1l=n−1Vl+∑θ=1θ=nWxn,θ,∪l=1l=n−2Ul+∑θ=1θ=nWxn,θ,∪l=1l=n−3Wl+∑θ=1θ=nWxn,θ.
We compute each of the factors in equation (39) separately.
The first factor is computed with the help of following distances:For2≤θ≤n and 1≤m≤θ−1,(40)dxn,θ,xl,m=n−l,form≤l≤n+m−θ−1,θ−m,forn+m−θ≤l≤n−1.For1≤θ≤n and θ≤m≤n,(41)dxn,θ,xl,m=n+m−l−θform≤l≤n−1.Thus, we get(42)∑θ=1θ=nWxn,θ,∪l=1l=n−1Vl=∑θ=1θ=n∑m=1m=θ−1∑l=ml=n−1dxn,θ,xlm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n−1dxn,θ,xlm+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1dxn,θ,xlm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n+m−θ−1dxn,θ,xlm+∑θ=2θ=n∑m=1m=θ−1∑l=n+m−θl=n−1dxn,θ,xlm+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1dxn,θ,xlm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n+m−θ−1n−l+∑θ=2θ=n∑m=1m=θ−1∑l=n+m−θl=n−1θ−m+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1n+m−l−θ=112n4−2n3−n2+2n+112n4−n2+112n4−n2=1123n4−2n3−3n2+2n.The second factor is computed with the help of following distances:For2≤θ≤n and 1≤m≤θ−1,(43)dxn,θ,ul,m=n−l,form≤l≤n+m−θ−1,θ−m,forn+m−θ≤l≤n−1.For1≤θ≤n and θ≤m≤n,(44)dxn,θ,ul,m=n+m−l−θ,form≤l≤n−1,∑θ=1θ=nWxn,θ,∪l=1l=n−1Ul=∑θ=1θ=n∑m=1m=θ−1∑l=ml=n−1dxn,θ,ulm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n−1dxn,θ,ulm+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1dxn,θ,ulm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n+m−θ−1dxn,θ,ulm+∑θ=2θ=n∑m=1m=θ−1∑l=n+m−θl=n−1dxn,θ,ulm+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1dxn,θ,ulm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n+m−θ−1n−l+∑θ=2θ=n∑m=1m=θ−1∑l=n+m−θl=n−1θ−m+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1n+m−l−θ=112n4−2n3−n2+2n+112n4−n2+112n4−n2=1123n4−2n3−3n2+2n.The third factor is computed with the help of following distances:For2≤θ≤n−1 and 1≤m≤θ−1,(45)dxn,θ,wl,m=n−l+1,form≤l≤n+m−θ−2,θ−m,forn+m−θ−1≤l≤n−2.Forθ=n and 1≤m≤θ−2,(46)dxn,θ,wl,m=n−mform≤l≤n−2.For1≤θ≤n−2 and θ≤m≤n−2,(47)dxn,θ,wl,m=n+m−l−θform≤l≤n−2,∑θ=1θ=nWxn,θ,∪l=1l=n−1Wl=∑θ=1θ=n∑m=1m=θ−1∑l=ml=n−1dxn,θ,wlm=∑θ=2θ=n∑m=1m=θ−1∑l=ml=n−1dxn,θ,wlm+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−1dxn,θ,wlm=∑θ=2θ=n−1∑m=1m=θ−1∑l=ml=n+m−θ−2dxn,θ,wlm+∑θ=2θ=n−1∑m=1m=θ−1∑l=n+m−θ−1l=n−2dxn,θ,wlm+∑θ=1θ=n−2∑m=θm=n−2∑l=ml=n−2dxn,θ,wlm+∑m=1m=n−2∑l=ml=n−2dxnn,wlm=∑θ=2−1θ=n∑m=1m=θ−1∑l=ml=n+m−θ−2n−l+1+∑θ=2θ=n−1∑m=1m=θ−1∑l=n+m−θ−1l=n−2θ−m+∑m=1m=n−2∑l=ml=n−2n−m+∑θ=1θ=n−2∑m=θm=n−2∑l=ml=n−2n+m−l−θ=112n4−6n3+11n2−6n+112n4−4n3+5n2−2n+1124n3−12n2+8n+112n4−2n3−n2+2n=1123n4−8n3+3n2+2n.The fourth factor is(48)WVn=∑θ=1θ=nWxn,θ=∑l=1l=n−1∑m=1m=n−lm=16n3−n.Putting equations (42), (44), (47), and (48) in (39), we get(49)WVn,ETn′=1129n4−10n3−9n2+4n.LetWUn−1,ETn″ be the distance between the vertices of Un−1 to itself and from ETn−1, and Wn−2, where Un−1=un−1,1,un−1,2,un−1,3,…,un−1,n−1, is(50)WUn−1,ETn″=∑θ=1θ=n−1Wun−1,θ,ETn−1+∑θ=1θ=n−1Wun−1,θ,Wn−2+WUn−1,WUn−1,ETn″=∑θ=1θ=n−1Wun−1,θ,∪l=1l=n−2Wl+∑θ=1θ=nWun−1,θ,∪l=1l=n−1Vl+∑θ=1θ=nWun−1,θ,∪l=1l=n−3Ul+∑θ=1θ=n−1Wun−1,θ.Again, we compute each of the factors separately.The first factor is computed with the help of following distances:For2≤θ≤n−2 and 1≤m≤θ−1,(51)dun,θ,wl,m=n−l−1,form≤l≤n+m−θ−2,θ−m+1,forn+m−θ−1≤l≤n−2.For1≤θ≤n−1 and θ≤m≤n−2,(52)dun,θ,wl,m=n+m−l−θform≤l≤n−2,∑θ=1θ=n−1Wun−1,θ,∪l=1l=n−2Wl=∑θ=1θ=n−1∑m=1m=n−1∑l=ml=n−1dun−1,θ,wlm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n−1dun−1,θ,wlm+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−2dun−1,θ,wlm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n+m−θ−2dun−1,θ,wlm+∑θ=2θ=n−2∑m=1m=θ−1∑l=n+m−θ−1l=n−2dun−1,θ,wlm+∑θ=1θ=n−1∑m=θm=n−2∑l=ml=n−2dun−1,θ,wlm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n+m−θ−2n−l−1+∑θ=2θ=n−2∑m=1m=θ−1∑l=n+m−θ−1l=n−2θ−m+1+∑θ=1θ=n−1∑m=θm=n−2∑l=ml=n−2n+m−l−θ=112n4−6n3+11n2−6n+112n4−2n3−n2+2n+112n4−2n3−n2+2n=1123n4−10n3+9n2−2n.The second factor is computed with the help of following distances:For2≤θ≤n−2 and 1≤m≤θ−1,(53)dun,θ,wl,m=n−l,form≤l≤n+m−θ−2,θ−m+1,forn+m−θ−1≤l≤n−2.For1≤θ≤n−2 and θ≤m≤n−2,(54)dun,θ,wl,m=n+m−l−θ,form≤l≤n−2,∑θ=1θ=n−1Wun−1,θ,∪l=1l=n−2Ul=∑θ=1θ=n−1∑m=1m=n−1∑l=ml=n−1dun−1,θ,ulm=∑θ=2θ=n−1∑m=1m=θ−1∑l=ml=n−2dun−1,θ,ulm+∑θ=1θ=n∑m=θm=n−1∑l=ml=n−2dun−1,θ,ulm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n+m−θ−2dun−1,θ,ulm+∑θ=2θ=n−1∑m=1m=θ−1∑l=n+m−θ−1l=n−2dun−1,θ,ulm+∑θ=1θ=n−1∑m=θm=n−2∑l=ml=n−2dun−1,θ,ulm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n+m−θ−2n−l+∑θ=2θ=n−2∑m=1m=θ−1∑l=n+m−θ−1l=n−2θ−m+1+∑θ=1θ=n−1∑m=θm=n−2∑l=ml=n−2n−l+j−θ=112n4−4n3−n2−12+112n4−2n3−n2+2n+112n4−2n3−n2+2n=1123n4−8n3−3n2+20n−12.The third factor is computed with the help of following distances:For2≤θ≤n−1 and 1≤m≤θ−1,(55)dun,θ,wl,m=n−l,form≤l≤n+m−θ−1,θ−m+1,forn+m−θ≤l≤n−1.For1≤θ≤n−2 and θ≤m≤n−2,(56)dun,θ,wl,m=n+m−l−θform≤l≤n−1,∑θ=1θ=n−1Wun−1,θ,∪l=1l=n−1Vl=∑θ=1θ=n−1∑m=1m=n−1∑l=ml=n−1dun−1,θ,xlm=∑θ=2θ=n−1∑m=1m=θ−1∑l=ml=n−1dun−1,θ,xlm+∑θ=1θ=n−1∑m=θm=n−1∑l=ml=n−1dun−1,θ,xlm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n+m−θ−2dun−1,θ,xlm+∑θ=2θ=n−1∑m=1m=θ−1∑l=n+m−θl=n−1dun−1,θ,xlm+∑θ=1θ=n−1∑m=θm=n−1∑l=ml=n−1dun−1,θ,xlm=∑θ=2θ=n−1∑m=1m=θ−1∑l=ml=n+m−θ−1n−l+∑θ=2θ=n−1∑m=1m=θ−1∑l=n+m−θl=n−1θ−m+1+∑θ=1θ=n−1∑m=θm=n−1∑l=ml=n−1n−l+j−θ=112n4−2n3−n2+2n+112n4−2n3−n2+2n+112n4−2n2=1123n4−4n3−3n2+4n.The fourth factor is(57)WUn−1=∑θ=1θ=n−1Wun−1,θ=∑l=1l=n−2∑m=2m=n−lm=1122n3−14n+12.
Putting equations (52), (54), (56), and (57) in (50), we get(58)WUn−1,ETn″=1129n9−20n3+3n2+8n.
Now, letWWn−2,ETn′″ be the distance between the vertices of Wn−2 to itself and from ETn−1, where Wn−2=wn−2,1,wn−2,2,wn−2,3,…,wn−2,n−2; then, WWn−2,ETn′″=∑θ=1θ=n−2Wwn−2,θ,ETn−1+Wn−2.
This is equivalent to(59)Wwn−2,ETn″′=∑θ=1θ=n−2Wwn−2,θ,∪l=1l=n−1Vl+∑θ=1θ=nWwn−2,θ,∪l=1l=n−1Ul+∑θ=1θ=nWwn−2,θ,∪l=1l=n−3Wl+∑θ=1θ=n−2Wwn−1,θ.
Again, we find each factor separately.The first factor is computed with the help of following distances:For1≤θ≤n−2 and 1≤m≤θ,(60)dun,θ,wl,m=n−l,form≤l≤n+m−θ−2,θ−m+1,forn+m−θ−1≤l≤n−1.For1≤θ≤n−2 and θ≤m≤n−2,(61)dun,θ,wl,m=n+m−l−θform≤l≤n−1,∑θ=1θ=n−2Wwn−2,θ,∪l=1l=n−1Vl=∑θ=1θ=n−2∑m=1m=n−1∑l=ml=n−1dwn−2,θ,xlm=∑θ=2θ=n−2∑m=1m=θ∑l=ml=n−1dwn−2,θ,xlm+∑θ=1θ=n−2∑m=θ+1m=n−1∑l=ml=n−1dwn−2,θ,xlm=∑θ=2θ=n−2∑m=1m=θ∑l=ml=n+m−θ−2dwn−2,θ,xlm+∑θ=2θ=n−2∑m=1m=θ∑l=n+m−θ−1l=n−1dwn−2,θ,xlm+∑θ=1θ=n−1∑m=θ+1m=n−1∑l=ml=n−1dwn−2,θ,xlm=∑θ=2θ=n−2∑m=1m=θ∑l=ml=n+m−θ−2n−l+∑θ=2θ=n−2∑m=1m=θ∑l=n+m−θ−1l=n−1θ−m+1+∑θ=1θ=n−1∑m=θ+1m=n−1∑l=ml=n−1n+m−l−θ−1=112n4−2n3−n2+2n+112n4−4n3+5n2−2n+112n4−4n3+5n2−2n=1123n4−10n3+9n2−2n.The second factor is computed with the help of following distances:For1≤θ≤n−2 and 1≤m≤θ−1,(62)dun,θ,wl,m=n−l,form≤l≤n+m−θ−2,θ−m+1,forn+m−θ−1≤l≤n−1.For1≤θ≤n−1,(63)dun,θ,wl,m=n−lforθ≤l≤n−2.For1≤θ≤n−2 and θ≤m≤n−2,(64)dun,θ,wl,m=n+m−l−θform≤l≤n−1,∑θ=1θ=n−2Wwn−2,θ,∪l=1l=n−2Ul=∑θ=1θ=n−2∑m=1m=n−2∑l=ml=n−2dwn−2,θ,ulm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n−2dwn−2,θ,ulm+∑θ=1θ=n−2∑l=θl=n−2dwn−2,θ,ulm+∑θ=1θ=n−2∑m=θ+1m=n−2∑l=ml=n−2dwn−2,θ,ulm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n+m−θ−2dwn−2,θ,ulm+∑θ=2θ=n−2∑m=1m=θ−1∑l=n+m−θ−1l=n−2dwn−2,θ,ulm+∑θ=1θ=n−2∑l=θl=n−2dwn−2,θ,ulm+∑θ=1θ=n−2∑m=θ+1m=n−2∑l=ml=n−2dwn−2,θ,ulm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n+m−θ−2n−l+∑θ=2θ=n−2∑m=1m=θ−1∑l=n+m−θ−1l=n−2θ−m+1+∑θ=1θ=n−2∑l=θl=n−2n−l+∑θ=1θ=n−2∑m=θ+1m=n−2∑l=ml=n−2n−l+m+θ−1=112n4−4n3−n2+16n−12+112n4−6n3+11n2−6n+1122n3−14n+12+112n4−6n3+11n2−6n=1123n4−14n3+21n2−10n.The third factor is computed with the help of following distances:For2≤θ≤n−2 and 1≤m≤θ−1,(65)dun,θ,wl,m=n−l−1,form≤l≤n+m−θ−3,θ−m−1,forn+m−θ−2≤l≤n−3.For1≤θ≤n−2 and θ≤m≤n−3,(66)dun,θ,wl,m=n+m−l−θ−1form≤l≤n−3,∑θ=1θ=n−2Wwn−2,θ,∪l=1l=n−2Wl=∑θ=1θ=n−2∑m=1m=n−3∑l=ml=n−3dwn−2,θ,wlm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n−3dwn−2,θ,wlm+∑θ=1θ=n−2∑m=θm=n−3∑l=ml=n−3dwn−2,θ,wlm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n+m−θ−3dwn−2,θ,wlm+∑θ=2θ=n−2∑m=1m=θ−1∑l=n+m−θ−2l=n−3dwn−2,θ,wlm+∑θ=1θ=n−2∑m=θm=n−3∑l=ml=n−3dwn−2,θ,wlm=∑θ=2θ=n−2∑m=1m=θ−1∑l=ml=n+m−θ−3n−l−1+∑θ=2θ=n−2∑m=1m=θ−1∑l=n+m−θ−2l=n−3θ−m−1+∑θ=1θ=n−2∑m=θm=n−3∑l=ml=n−3n+m−l−θ−1=112n4−8n3+17n2+2n−24+112n4−6n3+11n2−6n+1122n4−20n3+82n2−160n+120=1124n4−34n3+110n2−164n+96.The fourth factor is(67)WWn−2=∑θ=1θ=n−2Wwn−2,θ=∑l=2l=n−2∑m=2m=n−lm=1122n3−6n2−8n+24.
Putting equations (61), (64), (66), and (67) in (59), we get(68)Wwn−2,ETn″′=11210n4−56n3+134n2−184n+120.
Now, the Wiener index of the graphETn is WETn=WETn−1+WVn,ETn′+WUn−1,ETn″+WWn−2,ETn″′. Hence, by using (49), (58), and (68), we get WETn=WETn−1+1/1228n4−86n3+134n2−172n+120.


## 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 dimensionn is denoted by HXn (Figure 3). The graph contains 3n2−3n+1 vertices and 9n2−15n+6 edges, where n is the number of vertices on one side of the hexagon [5]. There is only one vertex v which is at distance n−1 from every other corner vertices. This vertex is said to be the center of HXn and is represented by O.Figure 3
Hexagonal networkHXn and its extension.Theorem 5.
WpHXn=27n2−81n+48.Proof.
In order to find the vertexPI index of HXn, 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 HXn to form triangular mesh networks T2n−1 and T2n−1′ of dimension 2n−1, respectively. Define the vertex set of T2n−1 and T2n−1′ as follows: Vl=xlm:1≤l≤2n−1,1≤m≤l and Vl′=vlm′:1≤l≤2n−1,1≤m≤l.
It is easy to see thatVT2n−1∩VT2n−1′=V2n−1 and dxlm,vlm′>3 for l<2n−3. This implies that the Wiener polarity index of HXn can now be written in the following form:(69)WpHXn=2WpT2n−1−WpHXn,Tn−1−WpTn−1+WpV2n−2,V2n−2′+WpV2n−2,V2n−3′+WpV2n−3,V2n−2′−WpV2n−1.
From Theorem1, we know that WpTn=9n2−5n+6/2. This implies that(70)WpT2n−1=18n2−63n+54.
Now, we calculate the termsWpV2n−2,V2n−2′, WpV2n−2,V2n−3′, and WpV2n−3,V2n−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, for every v∈V2n−2 and v′∈V2n−2′, v,v′|dv,v′=3=4n−8. Therefore,(71)WpV2n−2,Vn−2′=4n−8.
Similarly, for everyv∈V2n−2 and v′∈V2n−3′, we have v,v′|dv,v′=3=8n−14.
And, for everyv∈V2n−3 and v′∈V2n−2′, we have(72)v,v′|dv,v′=3=8n−14.
This implies that(73)WpV2n−2,V2n−3′=4n−8,WpV2n−2,V2n−2′=4n−8.
The termWpHXn,Tn−1 is the cardinality of set of vertices of Tn−1 that are connected through a path of length 3 from the vertices of HXn. It is easy to see that(74)WpHXn,Tn−1=∑l=nl=n+2WpVl,Vn−1+∑l=nl=n+1WpVl,Vn−2+WpVn,Vn−3=∑l=nl=n+2WpVl,Vl−3+∑l=nl=n+1WpVl,Vl−2+WpVn,Vn−1=∑l=nl=n+24l−12+∑l=nl=n+12l−6+2n−6=∑l=nl=n+24l−12+∑l=nl=n+12l−6+2n−6=18n−40.
The distance between the set of vertices of the setV2n−1 is equal to WpV2n−1 and it is easy to see that(75)WpV2n−1=2n−4.
Now, by replacing the values of all these factors in equation (69) and simplifying, we get(76)WpHXn=2WpT2n−1−WpHXn,Tn−1−WpTn−1+WpV2n−2,V2n−2′+WpV2n−2,V2n−3′+WpV2n−3,V2n−2′−WpV2n−1=27n2−81n+48.


## 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 networkTn for different values of n is shown in Table 1. The values show that the Wiener index increases rapidly compared to Wiener polarity index as n 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.Table 1
Comparison of Wiener and Wiener polarity indices of the graph of triangular mesh network.
nWpTnWTn2033039491595274596541089790226581354281918975211025212471113241973112405300271349544223145946333315702885331681912117317945162789181080215115191214280095201377359895Figure 4
Comparison of Wiener and Wiener polarity indices of graphTn.


### 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 networkETn for different values of n is shown in Table 2. The values show that the Wiener index increases rapidly compared to Wiener polarity index as n 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.Table 2
Comparison of Wiener and Wiener polarity indices of ETn.
Comparison of Wiener and Wiener polarity index ofETnNWpETTnWETTn4423395147111963152921754665228840130209119723890101617410401121006686712264610431313325515692114392722889115466232513616546045133817632161400418724582052219823210792172092821399407Figure 5
Comparison of Wiener and Wiener polarity index of the equilateral triangular tetra sheet.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.


## 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 networkTn for different values of n is shown in Table 1. The values show that the Wiener index increases rapidly compared to Wiener polarity index as n 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.Table 1
Comparison of Wiener and Wiener polarity indices of the graph of triangular mesh network.
nWpTnWTn2033039491595274596541089790226581354281918975211025212471113241973112405300271349544223145946333315702885331681912117317945162789181080215115191214280095201377359895Figure 4
Comparison of Wiener and Wiener polarity indices of graphTn.


## 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 networkETn for different values of n is shown in Table 2. The values show that the Wiener index increases rapidly compared to Wiener polarity index as n 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.Table 2
Comparison of Wiener and Wiener polarity indices of ETn.
Comparison of Wiener and Wiener polarity index ofETnNWpETTnWETTn4423395147111963152921754665228840130209119723890101617410401121006686712264610431313325515692114392722889115466232513616546045133817632161400418724582052219823210792172092821399407Figure 5
Comparison of Wiener and Wiener polarity index of the equilateral triangular tetra sheet.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.



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*Source: 2757925-2021-12-20.xml*