The algebra of flows in graphs
| Speaker: | David Wagner |
|---|---|
| Affiliation: | University of Waterloo |
| Room: | Mathematics and Computer Building (MC) 5426 |
Abstract:
We sketch the construction of a finitely generated graded abelian group $K^\cdot(X)$ associated to a graph $X$ which encodes Kirchhoff's Current Law on $X$ and all its contractions, in such a way that $\hom(K^1(X),R)$ is the familiar (real) cycle-space of $X$. There is a split "deletion-contraction" short exact sequence which shows that $K^\cdot(X)$ is torsion-free and that its Poincar\'e polynomial is a specialization of the Tutte polynomial of $X$. Functoriality of $K^\cdot$ implies a functorial coalgebra structure: dualizing, we obtain a $B$-algebra structure on $\hom(K^\cdot(X),B)$ for any commutative ring $B$, functorial in both arguments. This leads to some inequalities for the numbers $d_j(X)=$rank($K^j(X))$, and a nice presentation for $\hom(K^\cdot(X),Q)$ This all looks suspiciously like the homology and cohomology of some complex algebraic variety $X$ which is a contravariant functor of $X$. Wish I know what $X$ is.