A New Matrix-Tree Theorem

Details A New Matrix-Tree Theorem A New Matrix-Tree Theorem

2001-09-17

arxiv.org/abs/math/0109104v2

Mathematics

Combinatorics

minor changes, 29 pages, version accepted for publication in Int. Math. Res. Notices

Scientific paper

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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