Author’s abstract: 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 \(\operatorname{Hom}(K^j(X),B)\) 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, \(\operatorname{Hom}(K^1(X), \mathbb{R})\) is the familiar (real) “cycle-space” of \(X\). We show that \(K(X)\) is torsion-free and that its Poincaré polynomial is the specialization \(t^{n-k} T_X(1/t,1+t)\) of the Tutte polynomial of \(X\) (here \(X\) has \(n\) vertices and \(k\) components). Functoriality of \(K\) induces a functorial coalgebra structure of \(K(X)\); dualizing, for any ring \(B\) we obtain a functorial \(B\)-algebra structure on \(\operatorname{Hom}(K(X),B)\). 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 10 open problems.