Log-concavity of some independence polynomials via a partial ordering

Schwenk proved in 1981 that the edge independence polynomial of a graph is unimodal. It has been known since 1987 that the (vertex) independence polynomial of a graph need not be unimodal. Alavi et al. have asked whether the independence polynomial of a tree is unimodal. We apply some results on the log-concavity of combinations of log-concave sequences toward establishing the log-concavity (and, thus, the unimodality) of the independence numbers of some families of trees and related graphs.