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

TL;DR

This paper explores the properties of Speyer's matroid polynomial, establishing connections with graph connectivity, matroid invariants, and providing computational tools and data revealing new graph invariants and relationships.

Contribution

It introduces new relations between graph connectivity, matroid invariants, and Speyer's polynomial, along with an improved algorithm and data set for computing it.

Findings

Relation between $g_M'(-1)$ and graph properties

New constraints on $g_M''(-1)$ as a graph invariant

Proposed link between flow polynomial and $g_M''(0)$ for cubic graphs

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.