Abstract: In 1987, Alavi, Malde, Schwenk and Erd\H{o}s conjectured that the independence polynomial of any tree or forest is unimodal. Although many researchers have been attracted by it, it is still open. Inspired by this conjecture, in this paper, we prove that rooted products of some trees preserve real-rootedness of independence polynomials. In particular, we can obtain that their independence polynomials are unimodal and log-concave.

Key words: Independence polynomial , Unimodality , Claw-free graph