On the multiplicity of matching polynomial roots and $θ$-critical graphs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915520084705280 |
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| author | Xu, Leyou |
| author_facet | Xu, Leyou |
| contents | 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 $θ$, where the graphs attaining the bound are related to $θ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23842 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the multiplicity of matching polynomial roots and $θ$-critical graphs Xu, Leyou Combinatorics 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 $θ$, where the graphs attaining the bound are related to $θ$-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. |
| title | On the multiplicity of matching polynomial roots and $θ$-critical graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.23842 |