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arXiv · 2506.18788

Graph theoretic properties of Speyer's matroid polynomial $g_M(t)$

Erik Panzer
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Abstract

We prove relations between the number of $k$-connected components of a graph, Crapo's invariant $\beta(M)$ 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.

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Erik Panzer. 2025-06-23. Graph theoretic properties of Speyer's matroid polynomial $g_M(t)$. https://arxiv.org/abs/2506.18788

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