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

Generalising the Heilman-Lieb Theorem from statistical physics, Chudnovsky and Seymour [J. Combin. Theory Ser. B, 97(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 subdivded claw H and any $Δ$, there is an open set F ⊆ C containing [0, ∞) such that the independence polynomial of any H-free graph of maximum degree $Δ$ has all of its zeros outside of F. We also show that no such result can hold when H 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 H-free graphs of bounded degree when H is a subdivided claw. The statements of these results are more subtle, but are again best possible in various senses.