On Connectivity Polynomial of Graphs

In this paper we introduce a new graph polynomial, say connectivity polynomial. Let Formula: see text be a simple graph of order Formula: see text. The connectivity polynomial of Formula: see text, denoted by Formula: see text, is Formula: see text, where Formula: see text and Formula: see text (for Formula: see text) is the number of connected induced subgraphs of Formula: see text with order Formula: see text. We investigate some properties of the connectivity polynomial. We find the connectivity polynomial of disjoint union and join of graphs. Finally, we find the coefficients of this polynomial in terms of the vertex connectivity and the order of graphs.