On the roots of edge cover polynomials of graphs

Let G be a simple graph of order n and size m. An edge covering of the graph G is a set of edges such that every vertex of the graph is incident to at least one edge of the set. Let e(G,k) be the number of edge covering sets of G of size k. The edge cover polynomial of G is the polynomial E(G,x)=∑k=1me(G,k)xk. In this paper, we obtain some results on the roots of the edge cover polynomials. We show that for every graph G with no isolated vertex, all the roots of E(G,x) are in the ball {z∈C:|z|<(2+3)21+3≃5.099}. We prove that if every block of the graph G is K2 or a cycle, then all real roots of E(G,x) are in the interval (−4,0]. We also show that for every tree T of order n we have ξR(K1,n−1)≤ξR(T)≤ξR(Pn), where −ξR(T) is the smallest real root of E(T,x), and Pn,K1,n−1 are the path and the star of order n, respectively.