On the edge cover polynomial of a graph

Let G be a simple graph of order n and size m . An edge covering of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. In this paper we introduce a new graph polynomial. The edge cover polynomial of G is the polynomial E ( G , x ) = ∑ i = 1 m e ( G , i ) x i , where e ( G , i ) is the number of edge coverings of G of size i . Let G and H be two graphs of order n such that δ ( G ) ≥ n 2 , where δ ( G ) is the minimum degree of G . If E ( G , x ) = E ( H , x ) , then we show that the degree sequence of G and H are the same. We determine all graphs G for which E ( G , x ) = E ( P n , x ) , where P n is the path of order n . We show that if δ ( G ) ≥ 3 , then E ( G , x ) has at least one non-real root. Finally, we characterize all graphs whose edge cover polynomials have exactly one or two distinct roots and moreover we prove that these roots are contained in the set { − 3 , − 2 , − 1 , 0 } .