Lorentzian polynomials and log-concavity of the independence polynomials of graphs

Abstract

In this paper, we first construct two graphs $\mathcal{F}(l,m,t,s)$ and ${\mathcal{G}}_4(l,m,t,s)$. Then we introduce the graph ${\mathcal{F}}_n(l,m,t,s)$ and the operator $E_{{\mathcal{G}}_4(l,m,t,s)}$, where ${\mathcal{F}}_n(l,m,t,s)$ is defined by identifying the vertex $c$ of $n$ copies of $\mathcal{F}(l,m,t,s)$, and $E_{{\mathcal{G}}_4(l,m,t,s)}$ is defined by replacing each edge of $G$ with ${\mathcal{G}}_4(l,m,t,s)$, for any simple finite undirected graph $G$. By using the theory of Lorentzian polynomials, we prove that the independence polynomials of the graphs ${\mathcal{F}}_n(l,m,t,s)$ and the image graphs of $E_{{\mathcal{G}}_4(l,m,t,s)}$ are log-concave, respectively. As applications, our results not only make progress on the conjecture of Alavi, Malde, Schwenk and Erdős, but also generalize the results of Bendjeddou and Hardiman.