Listening to the cohomology of graphs

Abstract

We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we prove that L-L^(-1) is similar to the Hodge Laplacian H of G which is in one dimensions the direct sum of the Kirchhoff Laplacian H0 and its 1-form analog H1. The spectrum of a single choice of H0,H1 or H alone determines the Betti numbers b0,b1 of G as well as the spectrum of the other matrices. It follows that b0 is the number of eigenvalues 1 of L and that b1 is the number of eigenvalues -1 of L. For a general abstract finite simplicial complex G, we express the matrix entries g(x,y) = w(x) w(y) X( St(x) cap St(y) ) of the inverse of L using stars St(x)= { z in G | x subset of z } of x and w(x)=(-1)^dim(x) and Euler characteristic X. One can see W+(x)=St(x) and W-(x)={ z in G | z subset x } as stable and unstable manifolds of a simplex x in G and g(x,y) =w(x) w(y) X(W+(x) cap W+(y)) as heteroclinic intersection numbers or curvatures and the identity L g=1 as a collection of Gauss-Bonnet formulas. The homoclinic energy w(x)=X(W+(x) cap W-(x)) by definition adds up to X(G). The matrix M(x,y)=w(x) w(y) X(W-(x) cap W-(y)) is similar to L(x,y)=X(W-(x) cap W-(y)). The sum of the matrix entries of M is the definition of Wu characteristic. For dimension 2 and higher we don't know yet how to recover the Betti numbers from the eigenvalues of the matrix H or from L. So far, it can only be obtained from a collection of block matrices, via the Hodge relations b_k = dim(H_k). A natural conjecture is that for a Barycentric refinement of a complex G, the spectrum of L determines the Betti vector. We know this now in one dimensions.