We define a contravariant functor K from the category of finite graphs
and graph morphisms to the category of finitely generated graded abel
ian groups and homomorphisms. For a graph X, an abelian group B, and a
nonnegative integer j, an element of Hom(K-j(X), B) is a coherent fam
ily 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-1(X), R) is the
familiar (real) ''cycle-space'' of X. We show that K-.(X) is torsion-f
ree and that its Poincare polynomial is the specialization t(n-k)T(X)(
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 struc
ture on K-.(X); dualizing, for any ring B we obtain a functorial B-alg
ebra structure on Hom(K-.(X), B). When B is commutative we present thi
s algebra as a quotient of a divided power algebra, leading to some in
teresting inequalities on the coefficients of the above Poincare polyn
omial. We also provide a formula for the theta function of the lattice
of integer-valued flows in X, and conclude with 10 open problems. (C)
1998 Academic Press