# Enumerative Combinatorics of Graphs and Cell Complexes: Theorems of Trent and of Kirchhoff and Reidemeister-Franz Torsion

## Sylvain Cappell, Courant Institute

## October 27, 2015

Trentâ€™s theorem states that the determinant of the mesh matrix on
1-cycles of a connected graph is equal to the number of spanning
trees. In this talk this is extended to the mesh matrix on d cycles in
an arbitrary CW complex and even to the complete characteristic
polynomial of the mesh matrix on d cycles. This last is well defined
once a basis for the integral d-cycles is chosen. Even for graphs this
extension is of interest. Additionally, parallel theorems are given
for the mesh matrices on d boundaries. Naturally, the quotient of the
determinants for the d cycle by the d boundary mesh matrices records
the corresponding notion for the d homologies. A calculation of
Reidemeister-Franz torsion of the CW complex yields interesting
combinatorial relations among these and the product of the eigenvalues
of combinatorial Laplacians. These last had been given a combinatorial
description in work of Lyons and were studied by Cantanzaro, Chernyak
and Klein.

This is joint work with Edward Miller.