Around matrix-tree theorem

Abstract

Generalizing the classical matrix-tree theorem we provide a formula counting, for a given graph, its subgraphs with a fixed 2-core. We use this generalization to obtain an analog of the matrix-tree theorem for the root system Dn (the classical theorem corresponds to the A n-case). Several byproducts of the developed technique, such as a new formula for a specialization of the multivariate Tutte polynomial, are of independent interest. © International Press 2006.