A New Matrix-Tree Theorem [article]

Gregor Masbaum, Arkady Vaintrob
2002 arXiv   pre-print
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
more » ... prove some algebraic properties of our Pfaffian-tree polynomial.
arXiv:math/0109104v2 fatcat:v5re77m3qfby5czl4bjoaatwe4