Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs

Abstract

Let ${\text{Fl}}_{n,q}$ be the simplicial complex whose vertices are the non-trivial subspaces of ${\mathbb{F}}_q^n$ and whose simplices correspond to families of subspaces forming a flag. Let ${\Delta }_k^{+}({\text{Fl}}_{n,q})$ be the $k$-dimensional weighted upper Laplacian on ${\text{Fl}}_{n,q}$. The spectrum of ${\Delta }_k^{+}({\text{Fl}}_{n,q})$ was first studied by Garland, who obtained a lower bound on its non-zero eigenvalues. Here, we focus on the $k=0$ case. We determine the asymptotic behavior of the eigenvalues of ${\Delta }_0^{+}({\text{Fl}}_{n,q})$ as $q$ tends to infinity. In particular, we show that for large enough $q$, ${\Delta }_0^{+}({\text{Fl}}_{n,q})$ has exactly $\lfloor n^2/4\rfloor +2$ distinct eigenvalues, and that every eigenvalue $\lambda \ne 0,n-1$ of ${\Delta }_0^{+}({\text{Fl}}_{n,q})$ tends to $n-2$ as $q$ goes to infinity. This solves the $0$-dimensional case of a conjecture of Papikian.