On the edge cover polynomial of a graph

Let GG be a simple graph of order nn and size mm. 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 GG is the polynomial E(G,x)=∑i=1me(G,i)xi, where e(G,i)e(G,i) is the number of edge coverings of GG of size ii. Let GG and HH be two graphs of order nn such that δ(G)≥n2, where δ(G)δ(G) is the minimum degree of GG. If E(G,x)=E(H,x)E(G,x)=E(H,x), then we show that the degree sequence of GG and HH are the same. We determine all graphs GG for which E(G,x)=E(Pn,x)E(G,x)=E(Pn,x), where PnPn is the path of order nn. We show that if δ(G)≥3δ(G)≥3, then E(G,x)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}{−3,−2,−1,0}.