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. Here we introduce a new graph polynomial. The edge cover polynomial of G is the polynomial E(G, x) = ∑m i=1 e(G, i)x , where e(G, i) is the number of edge covering sets 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 show that cycles and complete bipartite graphs are determined by their edge cover polynomials. Also we determine all graphs G for which E(G, x) = E(Pn, x), where Pn is the path of order n. We show that if δ(G) ≥ 3, then E(G, x) has at least one non-real root. We prove that the real roots of edge cover polynomial of trees are dense in the interval [−4,0]. Finally, we characterize all graphs whose edge cover polynomials have exactly one or two distinct roots. In fact their roots are contained in {−3,−2,−1, 0}. 2010 AMS Subject Classification: 05C31; 05C70.