  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.
