The lattice of integer flows of a regular matroid

TL;DR

This paper generalizes the study of integer flow lattices from graphs to regular matroids, establishing a precise criterion for when two such lattices are isometric based on matroid minors.

Contribution

It extends the understanding of integer flow lattices from graphs to regular matroids and characterizes lattice isometry in terms of matroid minors.

Findings

Lattice isometry corresponds to minor isomorphism of regular matroids.

The lattice of integer flows of a regular matroid is fully characterized by its contracted minor.

The result explains previous observations about graph lattices and their isometries.

Abstract

For a finite multigraph G, let \Lambda(G) denote the lattice of integer flows of G -- this is a finitely generated free abelian group with an integer-valued positive definite bilinear form. Bacher, de la Harpe, and Nagnibeda show that if G and H are 2-isomorphic graphs then \Lambda(G) and \Lambda(H) are isometric, and remark that they were unable to find a pair of nonisomorphic 3-connected graphs for which the corresponding lattices are isometric. We explain this by examining the lattice \Lambda(M) of integer flows of any regular matroid M. Let M_\bullet be the minor of M obtained by contracting all co-loops. We show that \Lambda(M) and \Lambda(N) are isometric if and only if M_\bullet and N_\bullet are isomorphic.