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.

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 Ja-

hangir 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 In other words, and the Diameter D of Jahangir graph J_2,m is equal to D(J_2,m) = 4.

I. If, , we have two case for first sentences of the Schultz, Modified Schultz polynomials of J_2,m.

I-1. For a vertex, there are two path with length one until a vertex, thus there are 2m edges uvÎE(J_2,m), such that,. Therefore, we have two terms 5 × 2mx¹, 6 × 2mx¹ of the Schultz and Modified Schultz polynomials of Jahangir graph J_2,m, respectively.

I-2. For only vertex, there are m path with length one until a vertex, thus there are m edges, such that,. So, we have two terms and of the Schultz 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

and, respectively.

II. If, , we have three case for first sentences of the Schultz, Modified Schultz polynomials of J_2,m.

II-1. For a vertex, 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. Therefore, we have a terms 4 × mx² of the Schultz and Modified Schultz polynomials of J_2,m.

II-2. For every vertex, there are only 2-edge-path until the Center vertex c, and there are m

2-edge-path in J_2,m with and. Therefore, we have two terms, 2m × mx² of the Schultz and Modified Schultz polynomials of J_2,m, respectively.

II-3. For a vertex, there are m − 1 path with length two until other vertices, so there are 2-edge-path in J_2,m, such that,. So, we have two terms and 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 and respectively.

III. If, , for a vertex, there are (m − 2)m path with length three until vertices of B, such that,. Therefore, we have two sentences and of the Schultz and Modified Schultz polynomials of Jahangir graph J_2,m, respectively.

IV. If, , for a vertex, there are m − 3 path with length 4 = D(J_2,m), between v and other vertices u of A. Thus by, 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 Î ℕ − {2}.

and

And these complete the proof.

Theorem 2. Let J_2,m be the Jahangir graphs ("m ≥ 3). Then, the Schultz, Modified Schultz indices of J_2,m are equal to

Proof. 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 = 1). Thus we have following computations "m Î ℕ − {2}.

And

Here the proof of theorem is completed.

The author is thankful to Professor Emeric Deutsch from Department of Mathematics of Polytechnic University (Brooklyn, NY 11201, USA) for his precious support and suggestions.

Shaohui Wang,Mohammad Reza Farahani,M. R. Rajesh Kanna,R. Pradeep Kumar, (2016) Schultz Polynomials and Their Topological Indices of Jahangir Graphs J_2,m. Applied Mathematics,07,1632-1637. doi: 10.4236/am.2016.714140