On Laplacians and the disorientability of a simplicial complex

Abstract

Let K be an N-dimensional simplicial complex. We investigate the spectrum of the up Laplacian matrix of K. Let L be the \((N-1)\)th up Laplacian matrix of K. We show that the largest eigenvalues of L and |L| are equal if and only if K is disorientable. We also derive lower bounds for the sum of the first k largest eigenvalues of L.