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<p>In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal. This conjecture is still open. Recently, Zhu et al. confirmed this conjecture for certain class of trees. Motivated by them, in this paper, we first construct infinite families of trees and their composite graphs, and then obtain that their independence polynomials have only real zeros. In consequence, we also confirm the conjecture of Alavi et al. for more special cases. In fact, several results in this paper generalize those of Zhu et al.</p>

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Trees with Independence Polynomials Having Only Real Zeros

  • 期刊论文

  • 发表日期：

    2025年12月8日

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  • Lily Li Liu,
  • Hongyue Tang,
  • Jingrui Zhao

摘要

In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal. This conjecture is still open. Recently, Zhu et al. confirmed this conjecture for certain class of trees. Motivated by them, in this paper, we first construct infinite families of trees and their composite graphs, and then obtain that their independence polynomials have only real zeros. In consequence, we also confirm the conjecture of Alavi et al. for more special cases. In fact, several results in this paper generalize those of Zhu et al.