Schultz Polynomials and Their Topological Indices of Jahangir Graphs J 2 , m

Let G = (V; E) be a simple connected graph. The Wiener index ( ) ( ) { } ( ) ∑ v u V G W G d u v ∈ = , , is the sum of distances between all pairs of vertices of a connected graph. The Schultz topological index is equal to ( ) ( ) ( ) ( ) ∑ V u v G v u Sc G d d d u v ∈ = + , , 1 2 and the Modified Schultz topological index is ( ) ( ) ( ) ( ) ∑ u v u v V G Sc G d d d u v ∗ ∈ = × , , 1 2 . In this paper, the Schultz, Modified Schultz polynomials and their topological indices of Jahangir graphs J2,m for all integer number m ≥ 3 are calculated.


Introduction
Let G = (V; E) be an undirected connected graph without loops or multiple edges.The sets of vertices and edges of G are denoted by V(G) and E(G), respectively.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.An oldest topological index in chemistry is the Wiener index, that first introduced by Harold Wiener 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 a molecular graph G was defined as [1]: where the summation goes over all pairs of vertices of G and d(u, v) denotes the distance of the two vertices u and v in the graph G (the number of edges in a shortest path connecting u and v).For details of mathematical properties and applications, the readers are suggested to refer to [2]- [4] and the references therein.Other properties and applications of Wiener index can be found in [5]- [12].In 1989, H.P. Schultz [13] has introduced a graph theoretical descriptor for characterizing alkanes by an integer number as follow: where d u and d v are degrees of vertices u and v. Schultz named this descriptor the "molecular topological index" and denoted it by MTI.Later MTI became much better known under the name the Schultz index.
In 1997, S. Klavžar and I. Gutman [14] defined another based structure descriptors the Modified Schultz index of G is defined as: Now, there are two topological polynomials of a graph G as follow: and For more details about the Schultz, Modified Schultz polynomials and their topological indices and other molecular topological polynomials and indices reader can see the paper series [13]- [29].
In this paper we study the Schultz, Modified Schultz polynomials and their topological indices of Jahangir graphs J 2,m for all integer number m ≥ 3.

Main Results
In this section we compute the Schultz, Modified Schultz polynomials and their topological indices for Jahangir graphs J 2,m ∀m ≥ 3. The general form of Jahangir graphs J n,m is defined as follows: Definition 1. [30]- [35] Jahangir graphs J n,m for m ≥ 3, is a graph on nm + 1 vertices i.e., a graph consisting of a cycle C nm with one additional vertex which is adjacent to m vertices of C nm at distance n to each other on C nm .
Theorem 1.Let J 2,m be the Jahangir graphs (∀m ≥ 3).Then, The Schultz polynomial of J 2,m is equal to The Modified Schultz polynomial of J 2,m is equal to Proof.∀m ≥ 3 consider Jahangir graph J 2,m .By using Definition 1 and [29]- [32], one can see that the number of vertices in Jahangir graph J 2,m is equal to And the number of edges of Jahangir graph J 2,m is equal to ( ) Because, there is only Center vertex with degree m and there are m vertices with degree 2 and m vertices with degree.In this paper, we denote the sets of all vertices with degree two by A, all vertices with degree three by B and only Center vertex c by C. From the structure of Jahangir graph J 2,m (Figure 1), we see that there are distances from one to four, for every vertices , . , , there are m − 1 path with length two until other vertices ( ) of the Schultz and Modified Schultz polynomials of J 2,m , respectively.
Thus, the second sentences of the Schultz and Modified Schultz polynomials of Jahangir graph J 2,m are equal to ( ) , , ( ) , there are (m − 2)m path with length three until vertices of B, such that 5   ) And these complete the proof.
Here the proof of theorem is completed.  of distances between all pairs of vertices of a connected graph.The Schultz topological index

2 . 3 .
we have two case for first sentences of the Schultz, Modified Schultz po- lynomials of J 2,m .Therefore, we have two terms 5 × 2mx 1 , 6 × 2mx 1 of the Schultz and Modified Schultz polynomials of Jahangir graph J 2,m , respectively.I-For only vertex and Modified Schultz polynomials of J 2,m , respectively.Thus, the first sentences of the Schultz and Modified Schultz polynomials of Jahangir graph J 2,m are equal to = , we have three case for first sentences of the Schultz, Modified Schultz polynomials of J 2,m .II-1.For a vertex ( ) 2,m v A V J ∈ ⊂ , there are two path with length two until other vertices A, so there are (1/2) × 2m 2-edge-path in J 2,m , such that 4 u v u v d d d d + = × = .Therefore, we have a terms 4 × mx 2 of the Schultz and Modified Schultz polynomials of J 2,m .II-2.For every vertex ( ) 2,m v A V J ∈ ⊂ , there are only 2-edge-path until the Center vertex c, and there are m 2-edge-path in J 2,m with 2 mx 2 of the Schultz and Modified Schultz polynomials of J 2,m , respectively.II-For a vertex ( ) the Schultz and Modified Schultz polynomials of Jahangir graph J 2,m , respectively.m − 3 path with length 4 = D(J 2,m ), between v and other vertices u of A. Thus by 4 d + = × = , the fourth sentence of the Schultz and Modified Schultz polynomials of Jahangir graph J 2,m is equal to From the definition of the Schultz, Modified Schultz polynomials and above mentions, we have following results ∀m ∈ ℕ − {2}.

 Theorem 2 .
Let J 2,m be the Jahangir (∀m ≥ 3).Then, the Schultz, Modified Schultz indices of J 2,m are equal to Consider the Jahangir graph J 2,m (∀m ≥ 3) that presented in above proof.Now, by using the results from proof of Theorem 1 and according to the definitions of the Schultz, Modified Schultz indices of the graph G, one can see that these indices are the first derivative of their polynomials (evaluated at x = Thus we have following computations ∈ − {2}.