A New Matrix-Tree Theorem

TL;DR

This paper extends the classical Matrix-Tree Theorem to three-graphs, showing that their spanning trees can be generated by a Pfaffian, with implications for topological invariants like the Alexander-Conway polynomial.

Contribution

It introduces a Pfaffian-based method for enumerating spanning trees in three-graphs, generalizing the classical theorem and connecting to topological link invariants.

Findings

Spanning trees of three-graphs are generated by a Pfaffian.

The Pfaffian-tree polynomial has specific algebraic properties.

Topological interpretation relates to the Alexander-Conway polynomial.

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.