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 an on x n gives the number of independent subsets S ⊆ V (G) of vertices of G such that |S| = n. I(G, x) is unimodal if there is an index μ such that a0 ≤ a1 ≤ · · · ≤ aμ−1 ≤ aμ ≥ aμ+1 ≥ · · · ≥ ad−1 ≥ ad. 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.