Zero‐free regions for the independence polynomial on restricted graph classes

Abstract Generalising the Heilmann–Lieb theorem from statistical physics, Chudnovsky and Seymour [ J. Combin. Theory Ser. B , 97 (2007), no. 3, 350–357] showed that the univariate independence polynomial of any claw‐free graph has all of its zeros on the negative real line. In this paper, we show that for any fixed subdivided claw and any , there is an open set containing such that the independence polynomial of any ‐free graph of maximum degree has all of its zeros outside of . We also show that no such result can hold when is any graph other than a subdivided claw or if we drop the maximum degree condition. We also establish zero‐free regions for the multivariate independence polynomial of ‐free graphs of bounded degree when is a subdivided claw. The statements of these results are more subtle, but are again best possible in various senses.