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.