Independence polynomials of graphs with given cover number or dominate number

Abstract

For a graph [Formula: see text], let [Formula: see text] be the number of independent sets of size [Formula: see text] in [Formula: see text]. The independence polynomial [Formula: see text] has been the focus of considerable research. In this paper, using the coefficients of independence polynomials, we order graphs with some given parameters. We first determine the extremal graph whose all coefficients of [Formula: see text] are the largest (respectively, smallest) among all connected graphs (respectively, bipartite graphs) with given vertex cover number. Then we also derive the extremal graph whose all coefficients of [Formula: see text] are the largest among all connected graphs (respectively, bipartite graphs) with given vertex dominate number.