Graph theoretic properties of Speyer's matroid polynomial $g_M(t)$
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916807418314752 |
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| author | Panzer, Erik |
| author_facet | Panzer, Erik |
| contents | We prove relations between the number of $k$-connected components of a graph, Crapo's invariant $β(M)$ of a matroid, and Speyer's polynomial $g_M(t)$. These yield a simple interpretation of $g_M'(-1)$ when $M$ is graphic or cographic. Furthermore, we improve Ferroni's algorithm to compute $g_M(t)$ and provide an implementation and an extensive data set. These calculations reveal a large number of graph theoretic constraints on the second derivative $g_M''(-1)$, which we thus advertise as an intriguing new invariant of graphs. We also propose a relation between the flow polynomial and $g_M''(0)$ for cubic graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18788 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graph theoretic properties of Speyer's matroid polynomial $g_M(t)$ Panzer, Erik Combinatorics We prove relations between the number of $k$-connected components of a graph, Crapo's invariant $β(M)$ of a matroid, and Speyer's polynomial $g_M(t)$. These yield a simple interpretation of $g_M'(-1)$ when $M$ is graphic or cographic. Furthermore, we improve Ferroni's algorithm to compute $g_M(t)$ and provide an implementation and an extensive data set. These calculations reveal a large number of graph theoretic constraints on the second derivative $g_M''(-1)$, which we thus advertise as an intriguing new invariant of graphs. We also propose a relation between the flow polynomial and $g_M''(0)$ for cubic graphs. |
| title | Graph theoretic properties of Speyer's matroid polynomial $g_M(t)$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.18788 |