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 = 1 m e ( G , k ) x k . 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 ) 2 1 + 3 ≃ 5.099 } . We prove that if every block of the graph G is K 2 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 ξ R ( K 1 , n − 1 ) ≤ ξ R ( T ) ≤ ξ R ( P n ) , where − ξ R ( T ) is the smallest real root of E ( T , x ) , and P n , K 1 , n − 1 are the path and the star of order n , respectively.