Unimodality of the independence polynomials of non-regular caterpillars

The independence polynomial $I(G,x)$ of a graph $G$ is the polynomial in variable $x$ in which the coefficient $a_n$ on $x^n$ gives the number of independent subsets $S\subseteq V(G)$ of vertices of $G$ such that $\mathrm{\mid }S\mathrm{\mid }=n$. $I(G,x)$ is unimodal if there is an index $\mu$ such that that $a_0 \leq a_1 \leq$\mathrm{.}\mathrm{.}\mathrm{.}$\leq a_{\mu-1} \leq a_{\mu} \geq a_{\mu +1} \geq$\mathrm{.}\mathrm{.}\mathrm{.}$\geq a_{d-1} \geq a_d$ While the independence polynomials of many families of graphs with highly regular structure are known to be unimodal, little is known about less regularly structured graphs. We analyze the independence polynomials of a large infinite family of trees without regular structure and show that these polynomials are unimodal through a combinatorial analysis of the polynomials coefficients.