The Wiener Index and the Hosoya Polynomial of the Jahangir Graphs

Let G be a simple connected graph having vertex set V and edge set E. The vertex-set and edge-set of G denoted by V(G) and E(G), respectively. The length of the smallest path between two vertices is called the distance. Mathematical chemistry is the area of research engaged in new application of mathematics in chemistry. In mathematics chemistry, we have many topological indices for any molecular graph, that they are invariant on the graph automorphism. In this research paper, we computing the Wiener index and the Hosoya polynomial of the Jahangir graphs $J_5,m$ for all integer number $m \geq 3$.


Introduction
Let G be a connected graph. The vertex-set and edge-set of G denoted by V(G) and E(G), respectively. The degree of a vertex v  V(G) is the number of vertices joining to v and denoted by dv Topological indices of a simple graph are numerical descriptors that are derived from graph of chemical compounds. Such indices based on the distances in graph are widely used for establishing relationships between the structure of molecular graphs and Nanotubes and their physicochemical properties. Usage of topological indices in biology and chemistry began in 1947 when chemist Harold Wiener [1] introduced the Wiener index to demonstrate correlations between physicochemical properties of organic compounds and the index of their molecular graphs. Wiener originally defined his index on trees and studied its use for correlations of physico-chemical properties of alkanes, alcohols, amines and their analogous compounds [2] as: where d(u,v) denotes the distance between vertices u and v. The Hosoya polynomial of a graph is a generating function about distance distributing, introduced by Haruo Hosoya in 1988 and for a connected graph G is defined as [3]: In a series of papers, the Wiener index and the Hosoya polynomial of some molecular graphs and Nanotubes are computed. For more details about the Wiener index and the Hosoya polynomial, please see the paper series [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and the references therein.
In this research paper, we present some properties of the Wiener index and the Hosoya polynomial and we introduce a closed formula of this and the correspondent polynomial of the Jahangir graphs J5,m for all integer number m≥3.

Materials and Methods
The In the this section, we present some studies about a semi-regular and connected graphs that named "Jahangir Graphs Jn,m"  n≥2,  m≥3. And we introduce a method to compute the Wiener index and the Hosoya polynomial of the Jahangir graphs J5,m in continue.
What is a the Jahangir graphs Jn,m? Definition 2.1. [23,24] The Jahangir graphs Jn,m is a graph on nm+1 vertices and m(n+1) edges  n≥2 &  m≥3; i.e., a graph consisting of a cycle Cnm with one additional vertex which is adjacent to m vertices of Cnm at distance n to each other on Cnm. In particular,  m≥3 the Jahangir graphs J5,m is a graph consisting of a cycle C5m with one additional vertex (the Center vertex c) which is adjacent to m vertices of Cnm at distance 5 to each other on C5m. Some first example of this graphs are shown in Figure 1. For more details about the Jahangir graphs Jn,m reader can see the paper series [23][24][25][26][27][28][29][30][31][32][33][34][35][36].

Main Results and Discussion
In this paper, we computed a closed formula of the Wiener index and the Hosoya polynomial of the Jahangir graphs J5,m for all integer number m≥3.
Theorem 2.1. The Hosoya polynomial and the Wiener index for the Jahangir graphs J5,m for all integer number m≥3 are equal to 6 (3) Proof. Let J5,m be Jahangir graphs  m≥3 with 5m+1 vertices and 6m edges. From Definition 2.1 and Figure 1 V2={vV(J5,m)| dv=2} → |V2|=4m (5) V3={vV(J5,m)| dv=3} → |V3|=m (6) Vm={cV(J5,m)| dc=m} → |Vm|=1 (7) And alternatively, V(J5,m)=V2 V3Vm and V2 ∩V3∩Vm=∅ and we know where δ and Δ are the minimum and maximum of dv for all vV(G), respectively, thus Now, for compute the Hosoya polynomial and the Wiener index of J5,m, we denote the number of unordered pairs of vertices u and v of a graph G as distance d(u,v)=k by d(G,k) for all integer number k up to d(G) (where d(G) denote the topological diameter and is the longest distance between vertices of a graph G). Thus, we redefine this mention topological polynomial and index of G as follow: From the Definition 2.1 and the structure of Jahangir graphs J5,m in Figure 1 Because, there are 2|V3| 3-edges paths between cVm and vertices of V2, but there are not any 3-edges paths between c and vertices of V3. Also, there are ½×2|V3|+2|V3|=3m 3edges paths between all vertices of u,v  V2  V (J5,m)). From Because, there isn't any 6-edges paths started from members of C and V3 to all other vertices of Jahangir graphs J5,m in Figure 1. But there are ½×|V2|(2|V2|-5)=m(2m-5) 6edges paths between all vertices V2. Thus, the 6th and last sentence of H(J5,m,x) is m(2m-5)x 6 . Now, Equations 12, 13,…,17 imply that the Hosoya polynomial of the Jahangir graph J5,m is equal to: Here the proof of theorem is completed.