Trees with Real Rooted Independence Polynomials

Let I(G; x) denote the independence polynomial of a graph G. Alavi, Malde, Schwenk and Erdős conjectured for any tree T that I(T; x) is unimodal. In this paper, we derive that rooted products of some trees preserve real-rootedness of independence polynomials. In consequence, we obtain more and more trees whose independence polynomials have only real zeros. This in particular implies that their independence polynomials are unimodal.