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

Sylvain Cappell, Courant Institute

October 27, 2015

Trent’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.