ON THE GRAOVAC-PISANSKI INDEX

The Graovac-Pisanski index ( GP index) is an algebraic approach for generalizing the Wiener index. In this paper, we co mpute the difference between the Wiener and GP indices for an infinite family of polyhedral graph s.


INTRODUCTION
The Wiener number is defined as the half sum of distances between all pairs of vertices in a molecular graph, see: WIENER, 1947; GUTMAN and ŠOLTÉS, 1991;KLAVŽAR and ŽEROVNIK, 1996;DOBRYNIN et al., 2001DOBRYNIN et al., , 2002;;VUKIČEVIĆ and GRAOVAC, 2004;KLAVŽAR and GUTMAN, 2006;WAGNER, 2006WAGNER, , 2010;;ZHANG et al., 2008ZHANG et al., , 2010 Let G be a group and Ω be a non-empty set.An action of G on Ω is a function φ:G × Ω → Ω where (g,x) ↦φ(g,x) that satisfies the following two properties (we denote φ(g,x) as x g ): α e = α for all α in Ω and (α g ) h = α gh for all g,h in G.The orbit of an element α ∈ Ω is denoted by α G and it is defined as the set of all α g , g G ∈ .The stabilizer of element α ∈ Ω is defined as A bijection f on the vertices of graph X is called an automorphism of X which preserves the edge set E. In other words, the bijection f on V(X) is an automorphism if e=uv is an edge, then ( ) is an edge of E in which the image of vertex u is denoted by ( ) f u .The set of all automorphisms of X s denoted by Aut(X) .It is not difficult to see that Aut(X) under the composition of mappings forms a group.This group acts transitively on the set of vertices, if for a pair of vertices such as u and v in V(X), there is an automorphism Aut(X) g ∈ such that ( ) .g u v = The modified Wiener index was introduced in 1991 by A. GRAOVAC and T. PISANSKI to count the symmetries of a graph, see [17].The modified Wiener index is also called Graovac-Pisanski index as suggested M. GHORBANi and S. KLAVŽAR in [16].Consider the graph X with automorphism group Aut(X) G = . Then the Graovac-Pisanski index of X is Theorem 1 (GRAOVAC and PISANSKI, 1991).Let X be a graph with automorphism group Aut( ) G X = and vertex set V(X).Let V1, V2, …,Vk be all orbits under the action G on V(X).Then The difference between Wiener and modified Wiener indices is defined in HAKIMI-NEZHAAD and GHORBANI (2014) as  2) that the difference number is closely related to the number of orbits of Aut(X).In other words, the difference number is equal with the Wiener number if the regarded graph is asymmetric (a graph without non-trivial symmetry element).It is not difficult to see that in this case the number of orbits is equal with the number of vertices., 2017] introduced some new classes of polyhedral graphs with tetragons, pentagons, heptagons and octagons.In this paper, we also introduce an infinite class of cubic polyhedral graphs with tetragons, pentagons and hexagons denoted by (4,5,6)-polyhedral graphs.This class of polyhedral graphs has exactly 16n+8 vertices, where n is an integer greater than or equal with 3and thus, we denote this new family of cubic polyhedral graphs by C16n+8, see Figures 1  and 2.  Let s, p, h, n and m be respectively the number of tetragons, pentagons, hexagons, carbon atoms and bonds between them, in a given (4,5,6)

MAIN RESULTS AND DISCUSSION
Next, consider the action of subgroup G1.Any symmetry of the polyhedral graph C56 which fixes vertex 1 must also fixes the opposite vertex 3. Then applying again orbit-stabilizer property states that where n + 1 is the number of layers of L. Proof.Suppose the vertices of the last layer are U = {u1, u2, …,u16}.Let tn be two times the Wiener index of graph L. A straightforward computation yields the recurrence ).
To compute the summation , \ ( , ) ∑ by using the symmetry of the graph L, we have This implies that tn+1 Proof.First we partite the vertices of graph into three subsets B, U and W, where B = {v1, v2, …, vr}, U = {u1, …, us} and W = {w1, …, wr} are respectively the set of vertices of the internal cap, the vertices of nanotube L and the vertices of outer cap, see Figure 5.The distance matrix D can be written as following block form: .
The entries of matrix U is computed in Theorem 4. It is easy to see that the Wiener index is equal to the half-sum of distances of the distance matrix D between all pairs of vertices.For given polyhedral graph 8 16 + n C , the matrix V is constant as shown in Figure 6.The summation of entries of matrix V is 696.Obviously, the distance matrices B, U, and W depend to the number of rows in the nanotube L.

CONCLUSION
In this paper, we introduced a new family of cubic polyhedral graphs and then we computed its Graovac-Pisanski index.We also computed the difference between Wiener and GP indices for this class of polyhedral graphs.

Acknowledgement
This research is partially supported by the Shahid Rajaee Teacher Training University under grant number 27774.
This group is considered as the symmetry group of many molecular graphs such as fullerenes and polyhedral graphs.This group is of order 2n with two generators of orders n and 2. The cyclic group n ¢ of order n is also a group with generator g in which n ¢ = {g, g 2 , ..., g n = 1}.Lemma 2. Let n = 3.The automorphism group of the graph C16n+8 is isomorphic with the group The polyhedral graph C16n+8, for n=3is depicted in Figure3.Let G=Aut(C16n+8).If α denotes the rotation of C16n+8 for 90 o and β,γ are two reflections over the central vertical lines, polyhedral graph.By Euler's formula,we have

v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v 12 v 14 v 15 v 16 w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 8 w 9 w 10 w 11 w 12 w 14 w 15 w 16 v 13 w 13 u 1 u 2 u 3 u 4 u 5 u 6 u 8 u 9 u 10 u 11 u 12 u 13 u 14 u 15 u 16 u 17 u 18 u 19 u 20 u 21 u 22
In other words, if wn and wn-1 are the Wiener indices of the polyhedral graphs