The lattice of integer flows of a regular matroid

For a finite multigraph G, let Λ ( 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 Λ ( G ) and Λ ( 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 Λ ( M ) of integer flows of any regular matroid M . Let M • be the minor of M obtained by contracting all co-loops. We show that Λ ( M ) and Λ ( N ) are isometric if and only if M • and N • are isomorphic.