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

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}(