A New Matrix-Tree Theorem
Combinatorics
2007-05-23 v2 Geometric Topology
Quantum Algebra
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.
Cite
@article{arxiv.math/0109104,
title = {A New Matrix-Tree Theorem},
author = {Gregor Masbaum and Arkady Vaintrob},
journal= {arXiv preprint arXiv:math/0109104},
year = {2007}
}
Comments
minor changes, 29 pages, version accepted for publication in Int. Math. Res. Notices