Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs

Abstract

AbstractLet be the simplicial complex whose vertices are the nontrivial subspaces of and whose simplices correspond to families of subspaces forming a flag. Let be the ‐dimensional weighted upper Laplacian on . The spectrum of was first studied by Garland, who obtained a lower bound on its nonzero eigenvalues. Here, we focus on the case. We determine the asymptotic behavior of the eigenvalues of as tends to infinity. In particular, we show that for large enough , has exactly distinct eigenvalues, and that every eigenvalue of tends to as goes to infinity. This solves the zero‐dimensional case of a conjecture of Papikian.