The results mentioned in the slides before are now written down. This document contains a proof of the energy relation . There are several reason for setting … ….
The energy theorem for simplicial complexes equipped with a complex energy comes with some surpises.
If a set of set is equipped with an energy function, one can define integer matrices for which the determinant, the eigenvalue signs are known. For constant energy the matrix is conjugated to its inverse and defines two isospectral multi-graphs.
The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex.