Around matrix-tree theorem
Yurii Burman, Boris Shapiro
Abstract
Generalizing the classical matrix-tree theorem we provide a formula counting subgraphs of a given graph 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 An-case). Several byproducts of the developed technique, such as a new formula for a specialization of the multivariate Tutte polynomial, are of independent interest.
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