On the maximal independence polynomial of certain graph configurations

In this paper, we investigate the maximal independence polynomials of some popular graph configurations. Through careful analysis, some of the polynomials under study are shown to be Chebyshev, which helps characterize polynomial properties such as unimodality, log-concavity and real-rootedness with ease and efficiency. We partially characterize the bridge path and bridge cycle graphs of wheels and fans according to their unimodality properties and propose relevant open problems. Also, to compare with the usual independence polynomials, we provide analogous results regarding the vertebrated graph, and the firecracker graph, as studied by Wang and Zhu [47].