On vertex PI index of certain triangular tessellation networks

: The Wiener index, due to its many applications is considered to be one of very important distance-based index. But the Padmaker-Ivan (PI) index is kind of the only distance related index linked to parallelism of edges. The PI index like other distance related indices has great disseminating power. The index was firstly investigated by Khadikar et al. (2001), they have probed the chemical applications of the PI index. They proved that the proposed PI index correlates highly with the physicochemical properties and biological activities of a large number of diverse and complex chemical compounds and the Wiener and Szeged indices. Recently, the vertex Padmarkar-Ivan (PI v ) index of a chemical graph G was introduced as the sum over all edges uv of a molecular graph G of the vertices of the graph that are not equidistant to the vertices u and v . In this paper, the vertex PI v index of certain triangular tessellation are computed by using graph-theoretic analysis, combinatorial computing, and edge-dividing technology.

molecular descriptor. Wiener applied this index to determine the physical properties of certain types of alkanes known as paraffins. The Wiener index, due to its many applications is considered to be one of very important distance based index. But, the Padmaker-Ivan (PI) index is kind of the only distance related index linked to parallelism of edges.
There are many degree based and distance based topological indices that are defined by the mathematician and a lots of work has been done in this regard. Topological properties of molecular graphs in this regard are explored. The details about the work done in this direction can be found in Ali et al. (2017), Ashrafi and Loghman (2008), Baig et al. (2015), Caporossi et al. (2003), Estarda et al. (1998), Gao et al. (2016), Ghorbani and Hosseinzadeh (2010), Graovoc et al. (2011), Gutman and Das (2004), Gutman and Trinajstc (1972), and Kartica et al. (2012).
By introducing these topological indices, the graph theory has provided the chemist with a variety of very useful tool to investigate the chemical properties of certain chemical networks. Formally, a topological index is a numeric quantity from the structural graph of a molecule. There are many applications of these indices which are found in chemistry, pharmaceutics, and biology. For a thorough survey on this topic, consult the work of Mansour and Schork (2009).
In Amić et al. (1998), the authors defined a new distance related molecular topological index and named it as Padmarker-Ivan index. The newly defined index was abbreviated as PI v . This newly defined index was a distances related index and does not coincide with Wiener index in general and in particular, for acyclic (trees) molecules. One important property of the purposed index PI v is that it simple in calculation and has similar impact as that of the Wiener index, for detail see Ashrafi and Loghman (2006).

Introduction
The first molecular topological index that used in chemistry was Wiener index. It was in 1947, when Harold Wiener introduced a topological descriptor, known as the Wiener index that later become one of most useful and popular between two vertices x and y and it is defined to be the length of the path that contain least edges between the vertices x and y. The distance of an edge ∈ = ( ) e uv E G to the vertex ∈ ( ) w V G is defines as the minimum of the distances of its ends vertices to w, that is: The vertices which have shorter distance from the u than to v of the edge e are denoted by n u (e | G) and similarly, the vertices which have shorter distance to the vertex v than to the vertex u are denoted by n v (e | G). Thus: Similarly, we can define n v (e | G). The vertex Padmarker-Ivan (PI v ) index (Mansour and Schork, 2009) of a graph G is defined as:

PI G n e G n e G
From definition, it is clear that the vertices lying at equal distance from u and v are not counted. These vertices are said to be parallel to e. Thus, we can write:

The vertex PI index of graph derived from hexagonal networks
The graph derived from hexagonal networks are finite subgraphs of the triangular grid. In this section, the PI index of graph of hexagonal network is computed. The graph of hexagonal network of dimension n is denoted by HX(n). The graph contain − + 2 3 3 1 n n vertices and − + 2 9 15 6 n n edges, where n is the number of vertices on one side of the hexagon (Ashrafi and Loghman, 2006). There is only one vertex v which has distance n -1 from every other corner vertices. This vertex is said to be the center of HX(n) and is represented by O. In this section, The vertex PI index of hexagonal network HX(n) will be computed. We assume that V = V(HX(n)) is the set of vertices of HX(x) and for every e = uv;  Manuel et al. (2008) proposed a co-ordinate system in which direction of each correspond to an axis denoted by X, Y, and Z and the angle between any two axes is 120. It is therefore enough to calculate | | e N for every ∈ e E. To calculate N e , we divide the hexagon into six equilateral triangles S i ;i = 1, 2, 3 and S' i ;i = 1, 2, 3. Let ( ) e k N x denote the number of vertices in the k th row of HX(n) which are equidistance to edge e. Consider top left triangle S 1 . Now, we see that there are three types of edges, that are, horizontal edges e x parallel to X-axis, edges e y parallel to Y-axis and edges e z parallel to Z-axis. It is easy to see that N e for edges parallel to X-axis and Z-axis in S 1 and S' 1 are same. Similarly, N e for edges parallel to Y-axis and Z-axis in S 2 and S' 2 are same. N e for edges parallel to X-axis and Y-axis in S 3 and S' 3 are same.
Now: E S e m l l n be the set of edges parallel to Y-axis in S 1 . Let, the number of vertices in p-th row is denoted by x p of the graph HX(n). Since the graph HX(n) has 2n -1 rows, therefore the number of vertices in the graph HX(n) which are equidistance to end vertices of edge y lm e are:  and      Case 2. When n is odd:     We conclude from Lemma 3.1, Lemma 3.2, and Corollary 3.1 that: 1 .     (2 2 ) 2 2 2 1 2 (2 1)(2 1) = ( 2 2 The number of vertices of G is: ( 1)( 1). V G mn m n (3)

Conclusion
In this paper, we have solved vertex PI Index of certain triangular tessellations networks. We have considered hexagonal networks, triangular grids and enhanced meshes and analytical closed from results for these triangular tessellations networks were obtained. These results will be useful to understand the molecular topology of these important classes of networks.