Stability of independence polynomials of spiders

Source abstract

For a graph $G$, let $i_k(G)$ denote the number of independent sets of cardinality $k$, and let $$I(G,z)=\sum_{k\ge0} i_k(G)z^k$$ be its independence polynomial. Following Brown and Cameron \cite{BrownCameron2018}, a graph is called stable if all zeros of its independence polynomial lie in the closed left half-plane. They proved that every star is stable, but also constructed nonstable trees. They then asked for a characterization of stable trees. In this paper, we extend and strengthen their result by proving that every spider, obtained from a star by arbitrary and possibly nonuniform subdivisions of its edges, has all its independence roots in the open left half-plane. Hence, every spider is stable.