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

2025/06/23 by Panzer, Erik
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.18788

Abstract

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.

Related

• The omega invariant of a matroid
• M-Polynomial and Degree-Based Topological Indices
• Matroid variant of Matiyasevich formula and its application
• A construction of the graphic matroid from the lattice of integer flows
• Congruence conditions, parcels, and Tutte polynomials of graphs and matroids