讲座内容：

Motivated by the conjecture of Alavi, Malde, Schwenk and Erdos,

there are families of graphs, whose independence polynomials is unimodal, and furthermore is real-rooted. For example, Chudnovsky and Seymour obtained that the independence polynomial of all claw-free graphs has only real roots. Then it is natural to construct graphs with claw having real-rooted independence polynomials. In this paper, following the idea of Zhu et al., we introduce infinite graphs based on the rooted-product, whose independence polynomials have only real roots. Our results not only make progress on the conjecture of Alavi, Malde, Schwenk and Erdos, but also generalize Zhu et al.'s results.