On trees with real-rooted independence polynomial

The independence polynomial of a graph $G$ is

$I(G,x)=\sum _{k\ge 0}{i}_k\left(G\right)x^k,$

where $i_k\left(G\right)$ denotes the number of independent sets of $G$ of size $k$ (note that $i_0\left(G\right)=1$). In this paper we show a new method to prove real-rootedness of the independence polynomials of certain families of trees.In particular we will give a new proof of the real-rootedness of the independence polynomials of centipedes (Zhu’s theorem), caterpillars (Wang and Zhu’s theorem), and we will prove a conjecture of Galvin and Hilyard about the real-rootedness of the independence polynomial of the so-called Fibonacci trees.