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A New Matrix-Tree Theorem

Gregor Masbaum·Arkady Vaintrob
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Abstract

The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have exactly three vertices) the spanning trees are generated by the Pfaffian of a suitably defined matrix. This result can be interpreted topologically as an expression for the lowest order term of the Alexander-Conway polynomial of an algebraically split link. We also prove some algebraic properties of our Pfaffian-tree polynomial.

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BibTeXRIS

Gregor Masbaum, Arkady Vaintrob. 2002-02-28. A New Matrix-Tree Theorem. https://arxiv.org/abs/math/0109104

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