The Algebra of Flows in Graphs

Abstract

We define a contravariant functorKfrom the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graphX, an abelian groupB, and a nonnegative integerj, an element of Hom(Kj(X),B) is a coherent family ofB-valued flows on the set of all graphs obtained by contracting some (j-1)-set of edges ofX; in particular, Hom(K1(X),R) is the familiar (real) "cycle-space" ofX. We show thatK·(X) is torsion-free and that its Poincaré polynomial is the specializationtn-kTX(1/t, 1+t) of the Tutte polynomial ofX(hereXhasnvertices andkcomponents). Functoriality ofK·induces a functorial coalgebra structure onK·(X); dualizing, for any ringBwe obtain a functorialB-algebra structure on Hom(K·(X),B). WhenBis 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 inX, and conclude with 10 open problems. © 1998 Academic Press.