The Schultz Index and Schultz Polynomial of the Jahangir Graphs J 5 , m

Let G be simple connected graph with the vertex and edge sets V(G) and E(G), respectively. The Schultz and Modified Schultz indices of a connected graph G are defined as ( ) ( ) ( ) ( ) ∑ , 2 , 1 u v V G u v Sc G d d d u v ∈ = + and ( ) ( ) ( ) ( ) ∑ , 2 , 1 u v u v V G u v Sc G d d d ∈ ∗ = × , where d(u, v) is the distance between vertices u and v; dv is the degree of vertex v of G. In this paper, computation of the Schultz and Modified Schultz indices of the Jahangir graphs J5,m is proposed.


Introduction
Let G be simple connected graph with the vertex set V(G) and the edge set E(G).For vertices u and v in V(G), we denote by d(u, v) the topological distance i.e., the number of edges on the shortest path, joining the two vertices of G.
A topological index is a numerical quantity derived in an unambiguous manner from the structure graph of a molecule.As a graph structural invariant, i.e. it does not depend on the labelling or the pictorial representation of a graph.Various topological indices usually reflect molecular size and shape.
As an oldest topological index in chemistry, the Wiener index was first introduced by Harold Wiener [1] in 1947 to study the boiling points of paraffin.It plays an important role in the so-called inverse structure-property relationship problems.The Wiener index of G is defined as [1]- [7]: The Hosoya polynomial was introduced by Haruo Hosoya, in 1988 [8] and defined as follows: ( ) The number of incident edges at vertex v is called degree of v and denoted by d v .
The Schultz index of a molecular graph G was introduced by Schultz [9] in 1989 for characterizing alkanes by an integer as follow: The Modified Schultz index of a graph G was introduced by S. Klavžar and I. Gutman in 1996 as follow [10]:

∑
Also the Schultz and Modified Schultz polynomials of G are defined as: where d u and d v are degrees of vertices u and v.The Schultz indices have been shown to be a useful molecular descriptors in the design of molecules with desired properties, reader can see the paper series [11]- [29].
In this paper computation of the Schultz and Modified Schultz indices of the Jahangir graphs J 5,m are proposed.The Jahangir graphs J 5,m 3 m ∀ ≥ is defined as a graph on 5m + 1 vertices and 6 m edges i.e., a graph consisting of a cycle C 5m with one additional vertex (Center vertex c) which is adjacent to m vertices of C 5m at distance 5 to each other on C 5m .Some example of the Jahangir graphs and the general form of this graph are shown in Figure 1 and Figure 2 and the paper series [30]- [35].

Results and Discussion
In this present section, we compute the Schultz and Modified Schultz indices and the Schultz and Modified Schultz polynomials of the Jahangir graphs The Schultz index and polynomial are equal to Proof.Let J 5,m be Jahangir graphs 3 m ∀ ≥ with 5m + 1 vertices and 6 m edges.From Figure 1 and Figure 2, we see that 4 m vertices of J 5,m have degree two and m vertices of J 5,m have degree three and one additional vertex (Center vertex) of J 5,m has degree m.Thus we have three partitions of the vertex set ( ) ( ) ( ) V m− 2-edges paths between all vertices of ( ) x polynomials and indices of J 5,m by using all cases of the ( ) From the definition of Schultz index and the Schultz Polynomial of G, we can compute the Schultz index of the Jahangir graph J 5,m by the first derivative of Schultz polynomial of J 5,m (evaluated at x = 1) as follow:
as. Theorem 1.Let J 5,m be the Jahangir graphs for all integer numbers 3 m ∀ ≥ .Then, the Schultz, Modified Schultz polynomials and indices are as: = .Thus, the first terms of the Schultz and Modified Schultz polynomials of J 5,m are equal to m V V J ⊂ .Thus, the second terms of the Schultz and Modified Schultz polynomials of 5,m J By using the definition of the Jahangir graphs and Figure1and Figure2, we can compute other terms of the Schultz and Modified Schultz polynomials of J 5,m .We compute and present all necessary results on based the degrees of d u & d v for all cases of ( ), d u v -edge-paths ( ) , 1, 2, , 6 d u v =  infollowing table.Now, we can compute all coefficients of the Schultz

∂
And from the first derivative of Schultz Modified polynomial of the Jahangir graph J 5,m (evaluated at x = 1), the Modified Schultz index of J 5,m is equal to:Here these completed the proof of Theorem 1. ■

Table 1 .
for compute the Schultz and Modified Schultz indices and the Schultz and Modified Schultz polynomials of the Jahangir graphs All cases of ( )  of the Jahangir graph J 5,m .