Journal of Prime Research in Mathematics

# Computation of hosaya polynomial, wiener and hyper wiener index of jahangir graph $$j_{6,m}$$

Mehdi Rezaei
Department of Mathematics, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran.
Department of Applied Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran.
Waqas Khalid
Department of Mathematics, COMSATS Institute of Information Technology, Attock Campus, Pakistan.
Abdul Qudair Baig
Department of Mathematics, COMSATS Institute of Information Technology, Attock Campus, Pakistan.

$$^{2}$$Corresponding Author: Mr_Farahani@Mathdep.iust.ac.ir

### Abstract

Let $$G = (V, E)$$ be a simple connected graph with vertex set $$V$$ and edge set $$E$$. For two vertices $$u$$ and $$v$$ in a graph $$G$$, the distance $$d(u, v)$$ is the shortest path between $$u$$ and $$v$$ in $$G$$. Graph theory has much advancements in the field of theoretical chemistry. Recently, chemical graph theory is becoming very popular among researchers because of its wide applications of mathematics in chemistry. One of the important distance based topological index is the Wiener index, defined as the sum of distances between all pairs of vertices of $$G$$, defined as $$W(G) = \sum_{ u,v∈V (G)} d(u, v)$$. The Hosaya polynomial is defined as $$H(G, x) =\sum _{u,v∈V (G)} x ^{d(u,v)}$$. The hyper Wiener index is defined as $$WW(G) =\sum_{u,v∈V (G)} d(u, v) + \frac{1}{2}\sum_{u,v∈V (G)}d^{2}(u, v)$$. In this paper, we study and compute Hosaya polynomial, Wiener index and hyper Wiener index for Jahangir graph $$J_{6,m}$$, $$m ≥ 3$$. Furthermore, we give exact values of these topological indices.

#### Keywords:

Topological descriptors, Distance, Hosaya polynomial, Wiener index, Hyper Wiener index, Jahangir graph $$J_{6,m}$$.