ECCENTRIC CONNECTIVITY POLYNOMIALS AND THEIR TOPOLOGICAL INDICES OF JAHANGIR GRAPHS

Let $G=(V, E)$ be a simple and connected graph. The degree of a vertex $u$ and its eccentricity of a graph $G$ is denoted as $d(u)$ and $e(u)$ respectively. The eccentric connectivity polynomial $\xi^c(G,x)$ of a graph $G$ is defined as $\xi^c(G,x) = \sum _{u\in V(G)} {d(u) x^{e(u)}}$ and the modified eccentric connectivity polynomial $\xi_c(G,x)$ of a graph $G$ is defined as $\xi_c(G, x) = \sum _{u\in V(G)} {M(u)x^{e(u)}}$, where $M(u) = \sum _{v\in N_{G}(u)} d(v)$ i.e., sum of the neighbouring vertices of $u \in V(G)$. The first derivative of these polynomials evaluated at $x=1$ generates eccentric connectivity index $\xi^c(G)$ defined as $\xi^c(G) = \sum _{u\in V(G)} {d(u)e(u)}$ and modified eccentric connectivity index $\xi_c(G)$ defined as $\xi_c(G) = \sum _{u\in V(G)} {M(u)e(u)}$ respectively. In this paper, we present the generalized results for eccentric connectivity polynomial, modified eccentric connectivity polynomial and their respective indices for Jahangir graph $J_{n,m}$ with $n\geq 2$ and $m\geq 3$.