The Algebra of Flows in Graphs

We define a contravariant functor K· from the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graph X, an abelian group B, and a nonnegative integer j, an element of Hom(K(X), B) is a coherent family of B-valued flows on the set of all graphs obtained by contracting some (j − 1)-set of edges of X; in particular, Hom(K(X),R) is the familiar (real) “cycle-space” of X. We show that K·(X) is torsion-free and that its Poincaré polynomial is the specialization tTX(1/t, 1+ t) of the Tutte polynomial of X (here X has n vertices and k components). Functoriality of K· induces a functorial coalgebra structure on K·(X); dualizing, for any ring B we obtain a functorial B-algebra structure on Hom(K·(X), B). When B is commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincaré polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows in X, and conclude with ten open problems. 0. Introduction. Two ideas for defining algebraic invariants of graphs have been particularly successful up to now: these are the spectral and the K-theoretic approaches. The spectral theory begins by associating a (usually Hermitian) matrix M with a graph X , and proceeds by relating combinatorial structure in X to the spectral decomposition of M [3, 6, 10, 13]. The K-theoretic approach is the theory of the Tutte polynomial and its many interesting specializations [3, 7, 19, 20, 23]. A third method which seems promising but has been relatively neglected is the categorical approach. This idea is to define a functor from the category of graphs and graph morphisms into some (algebraic, geometric, topological,...) category, and to use these other structures to analyze the category of graphs. The most notable example of this approach so far is Walker’s functorial setting [24] for Lovász’s proof [14] of the Kneser conjecture; see also Björner’s survey [5]. Here we define such a functor K · into the category of finitely generated graded abelian groups and homomorphisms. The definition of K (X) is a formalization of Kirchhoff’s First Law on X and all of its images under repeated contractions of edges; for this reason we refer to K (X) as the “Kirchhoff group” of X . Imagine that the edges of X represent pipes, all of the same cross-sectional area, which are full of water (an incompressible fluid) and connected at the vertices. A flow of water in this system is represented by assigning a (real) velocity parallel to each edge; 1991 Mathematics Subject Classification. 05C99, 05E99, 18B99.