On the multiplicity of matching polynomial roots and $\theta$-critical graphs

Leyou Xu

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arXiv · 2509.23842

On the multiplicity of matching polynomial roots and $\theta$-critical graphs

Leyou Xu

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Abstract

The matching polynomial of a graph encodes rich combinatorial information through its roots. We determine the maximum multiplicity of a non-zero matching polynomial root and characterize all graphs attaining the bound. We also generalize the result to any fixed $\theta$, where the graphs attaining the bound are related to $\theta$-critical graphs. Inspired by these graphs, we give a constructive answer to Godsil's question. Finally, we show the existence of $1$-critical tree of order $n$ for all $n\ge 9$ and $1$-critical graph of order $n$ for all $n\ge 5$, and describe a method to construct $1$-critical graphs from existing ones.

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Leyou Xu. 2025-09-28. On the multiplicity of matching polynomial roots and $\theta$-critical graphs. https://arxiv.org/abs/2509.23842

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