Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs

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