Introduction to hub polynomial of graphs

Abstract

In this paper we introduce hub polynomial of a connected graph $\mathrm{G}$. The hub polynomial of a connected graph $\mathrm{G}$ of order $\mathrm{n}$ is the polynomial $H_G(x)=\sum_{i=h(G)}^n h_{G, i} x^i$ where $h_{G, i}$ denotes the number of hub sets of $\mathrm{G}$ of cardinality $\mathrm{i}$ and $h(G)$ is the hub number of $G$. We obtain hub polynomial of some special classes of graphs.We study hub roots of some graph $G$. Also we obtain hub polynomial of join of two graphs. We define $\mathrm{H}$-unique graphs and obtain some family of $\mathrm{H}$-unique graphs.