Enumerative Combinatorics of Simplicial and Cell Complexes: Kirchhoff and Trent Type Theorems

TL;DR

This paper explores the combinatorial and geometric properties of matrices associated with graphs and cell complexes, linking their characteristic polynomial coefficients to enumeration of substructures and relating to torsion and foundational graph theory work.

Contribution

It introduces new relationships between matrices of cell complexes and combinatorial enumeration, extending classical graph theory results to higher dimensions.

Findings

Coefficients of characteristic polynomials relate to enumeration of subobjects.

Connections established between matrices and Reidemeister-Franz torsion.

Relations to foundational work of Lyons and Kalai are clarified.

Abstract

This paper considers three separate matrices associated to graphs and (each dimension of) cell complexes. It relates all the coefficients of their respective characteristic polynomials to the geometric and combinatorial enumeration of three kinds of subobjects. The matrices are: the mesh matrix for integral d-cycles of Trent, the mesh matrix for integral d-boundaries, and the Kirchhoff matrix, i.e., the combinatorial Laplacian, for integral (d-1)-chains. Relations to Reidemeister-Franz torsion are elucidated and relations to the foundational work of R. Lyons and G. Kalai.