The lattice of integer flows of a regular matroid

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.