The Hosoya polynomial, Wiener index, and Hyper-Wiener index of Jahangir Graph $J_{8,m}$

Abstract

Let $G(V,E)$ be a simple connected graph with vertex set $V$ and edge set $E$. The Wiener index in the graph is $W(G)=\sum _{\{u,v\}\subseteq V}d(u,v),$ where $d(u,v)$ is the distance between $u$ and $v$, and the Hosoya polynomial of $G$ is $H(G,x)=\sum _{\{u,v\}\subseteq V}{x}^{d(u,v)}.$ The hyper-Wiener index of $G$ is $WW(G)=\frac{1}{2}(W(G)+\sum _{\{u,v\}\subseteq V}{d}^2(u,v)).$ In this paper, we compute the Wiener index, Hosoya polynomial, and hyper-Wiener index of Jahangir graph $J_{8,m}$ for $m\ge 3$.