The lattice of integer flows of a regular matroid

Details The lattice of integer flows of a regular matroid The lattice of integer flows of a regular matroid

2009-08-27

arxiv.org/abs/0908.4071v2

Mathematics

Combinatorics

18 pages, no figures. Revised version to appear in J. Combin. Theory Series B

Scientific paper

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.

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