# The energy of a simplicial complex

## TL;DR

This paper introduces a matrix-based energy concept for simplicial complexes, linking the inverse matrix entries to topological invariants like Euler characteristic and eigenvalue counts.

## Contribution

It establishes a novel connection between the inverse of a simplicial complex's intersection matrix and topological invariants such as Euler characteristic and eigenvalue distribution.

## Key findings

- Total energy equals the Euler characteristic of the complex.
- Number of positive minus negative eigenvalues equals the Euler characteristic.
- The intersection matrix is always unimodular, with an integer inverse.

## Abstract

A finite abstract simplicial complex G defines a matrix L, where L(x,y)=1 if two simplicies x,y in G intersect and where L(x,y)=0 if they don't. This matrix is always unimodular so that the inverse g of L has integer entries g(x,y). In analogy to Laplacians on Euclidean spaces, these Green function entries define a potential energy between two simplices x,y. We prove that the total energy summing all matrix elements g(x,y) is equal to the Euler characteristic X(G) of G and that the number of positive minus the number of negative eigenvalues of L is equal to X(G).