The energy of a simplicial complex

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=L−1 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 E(G)=∑x,yg(x,y) is equal to the Euler characteristic χ(G) of G and that the number of positive minus the number of negative eigenvalues of L is equal to χ(G).