PreprintarXiv licencearXiv
Stability of independence polynomials of spiders
Abstract
For a graph G, let ik(G) denote the number of independent sets of cardinality k, and let I(G,z)=∑k≥0 ik(G)zk 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.
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BibTeX
@misc{zhang2026stability,
title = {{Stability of independence polynomials of spiders}},
author = {Lei Zhang and Jianhua Tu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04694},
eprint = {2609.04694},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04694},
}
RIS
TY - UNPB
TI - Stability of independence polynomials of spiders
AU - Lei Zhang
AU - Jianhua Tu
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04694
UR - https://arxiv.org/abs/2609.04694
AB - For a graph G, let i_k(G) denote the number of independent sets of cardinality k, and let I(G,z)=∑_k≥0 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.
ER -