# Category: simplicial complex

## A multi-particle energy theorem

A finite abstraact simplicial complex or a finite simple graph comes with a natural finite topological space. Some quantities like the Euler characteristic or the higher Wu characteristics are all topological invariants. One can also reformulate the Lefschetz fixed point theorem for continuous maps on finite topological spaces.

## More on Analytic Torsion

We report on some progress on analytic torsion A(G) for graphs. A(G) is a positive rational number attached to a network. We can compute it for contractible graphs or spheres.

## More on Ringed Complexes

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 things up more generally and there is also some mentioning in the article: allowing general rings and not just division algebras extends the …

## Complex energized complexes

The energy theorem for simplicial complexes equipped with a complex energy comes with some surpises.

## Energized Simplicial Complexes

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

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.