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

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 sub-objects. The matrices are: the mesh matrix for integral d-cycles, the mesh matrix for integral d-boundaries, and the Kirchhoff matrix, i.e., the combinatorial Laplacian, for integral $$(d-1)$$ -chains. 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 (Trent in Proc Nat Acad Sci USA 40:1004–1007, 1954; J Acoust Soc Am 27(3):500–527, 1955). Here this theorem is extended to the mesh matrix on d-cycles in an arbitrary cell complex and, new even for graphs, to enumerative combinatorial interpretation of all of the coefficients of its characteristic polynomial. This last is well defined once a basis for the integral d-cycles is chosen. Additionally, a parallel result for the mesh matrix for integral d-boundaries is proved. Kirchhoff’s classical theorem on graphs (Kirchhoff in Ann Phys Chem 72:497–508, 1847) states that the product of the non-zero eigenvalues of the Kirchhoff matrix, i.e., combinatorial Laplacian, for connected graphs equals n times the number of spanning trees, with n the number of vertices. Kalai (Isr J Math 45(4): 337–351, 1983) investigated counting spanning trees in some higher dimensional settings including simplices here weighting by order of torsion homology groups entered. Lyons has generalized Kirchhoff’s result on the product of the non-zero eigenvalues of the Kirchhoff or combinatorial Laplacian on $$(d-1)$$ -chains to cell complexes for $$d >1$$ (Lyons in J Topol Anal 1(2), 153–175, 2009; Lyons and Peres in Probability on trees and networks. Cambridge series in statistical and probabilistic mathematics, vol 42, Cambridge University Press, New York, 2016). The present analysis extends this to all coefficients of the complete characteristic polynomial. An evaluation of the Reidemeister–Franz torsion of the cell complex with respect to its integral basis gives relations between these combinatorial invariants.