The Schultz Index and Schultz Polynomial of the Jahangir Graphs * J *_{5, m } ()

Mohammad Reza Farahani^{1}, Wei Gao^{2}

^{1}Department of Applied Mathematics of Iran University of Science and Technology (IUST), Narmak, Iran.

^{2}School of Information Science and Technology, Yunnan Normal University, Kunming, China.

**DOI: **10.4236/am.2015.614204
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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 and , where *d* (*u*, *v*) is the distance between vertices *u* and *v *; *d*_{v} is the degree of vertex *v* of *G*. In this paper, computation of the Schultz and Modified Schultz indices of the Jahangir graphs *J*_{5,m} is proposed.

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Farahani, M. and Gao, W. (2015) The Schultz Index and Schultz Polynomial of the Jahangir Graphs * J *_{5, m }. *Applied Mathematics*, **6**, 2319-2325. doi: 10.4236/am.2015.614204.

Received 13 November 2015; accepted 28 December 2015; published 31 December 2015

1. 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} 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] .

Figure 1. Some examples of the Jahangir graphs J_{5,3}, J_{5,4}, J_{5,5}, J_{5,6} and J_{5,8}.

Figure 2. A general representation of the Jahangir graphs n = 5,.

2. 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 n = 5, as.

Theorem 1. Let J_{5,m} be the Jahangir graphs for all integer numbers. Then, the Schultz, Modified Schultz polynomials and indices are as:

The Schultz index and polynomial are equal to

・

・ .

The Modified Schultz index and polynomial are equal to:

・

・

Proof. Let J_{5,m} be Jahangir graphs 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 as follow

Obviously, and thus

Now, for compute the Schultz and Modified Schultz indices and the Schultz and Modified Schultz polynomials of the Jahangir graphs, we see that for all vertices u, v in and the diameter of the Jahangir graph J_{5,m} is equal to.

Now, we compute all cases of d(u,v)-edge-paths of J_{5,m} in Table 1.

Table 1. All cases of -edge-paths of the Jahangir graph J_{5,m}.

For example, in case; one can see that there are 1-edges paths between the vertex c and vertices from V_{3} (where). There exist two 1-edges paths starts every vertex until (where). There are 3 m 1-edges paths between two vertices (two adjacent vertices or edges), such that. Thus, the first terms of the Schultz and Modified Schultz polynomials of J_{5,m} are equal to and respectively.

Also, in case; there are two 2-edges paths between Center vertex

and other vertices of vertex set. 2-edges paths between all vertices of

and 2-edges paths start from vertices of until vertices of and. Thus, the second terms of the Schultz and Modified Schultz polynomials of are equal to and, respectively.

By using the definition of the Jahangir graphs and Figure 1 and Figure 2, 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 -edge-paths in following table.

Now, we can compute all coefficients of the Schultz and Modified Schultz polynomials and indices of J_{5,m} by using all cases of the -edge-paths of the Jahangir graph J_{5,m} in Table 1 and alternatively

^{ }

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:

^{ }

And also Modified Schultz polynomial of J_{5,m} is equal to

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. ■

Acknowledgements

The authors are thankful to Professor Emeric Deutsch from Department of Mathematics of Polytechnic University (Brooklyn, NY 11201, USA) for his precious support and suggestions. The research is also partially supported by NSFC (No. 11401519).

NOTES

^{*}Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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