Unimodality of independence polynomials of two family of trees

Abstract

In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomials of trees are unimodal. Subsequently, many researchers proposed strengthening this conjecture to log-concavity. In 2023, Kadrawi, Levit, Yosef, and Mizrachi discovered independence polynomials of trees of order 26 that are not log-concave, which led them to construct two infinite families of such polynomials, denoted by $T_{3,m,n}$ and $T_{3,m,n}^*$. In this paper, we show that these two infinite families also satisfy the unimodal conjecture raised by Alavi, Malde, Schwenk, and Erdős.

Figures (10)

• Figure 1.1: The graph $T_{3,m,n}$.
• Figure 1.2: The graph $T^{\ast}_{3,m,n}$.
• Figure 2.1: The path $P_4$ and the clan graph $P_4^{(2,0,1,3)}$
• Figure 3.1: Spider $S(2^n)$.
• Figure 3.2: Spider $S(1^k, 2^r)$.

Theorems & Definitions (123)

• Conjecture 1.1: AMSE87
• Theorem 1.2
• Theorem 1.3: KL25
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Sta98
• Theorem 2.2: LLYZ25
• Lemma 2.3
• proof
• Corollary 2.4: LLYZ25