Garland's method for token graphs

The k-th token graph of a graph $G=(V,E)$ is the graph $F_k\left(G\right)$ whose vertices are the k-subsets of V and whose edges are all pairs of k-subsets $A,B$ such that the symmetric difference of A and B forms an edge in G. Let $L\left(G\right)$ be the Laplacian matrix of G, and $L_k\left(G\right)$ be the Laplacian matrix of $F_k\left(G\right)$. It was shown by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martínez that for any graph G on n vertices and any $0\le \ell \le k\le \lfloor n/2\rfloor$, the spectrum of $L_{\ell }\left(G\right)$ is contained in that of $L_k\left(G\right)$.

Here, we continue to study the relation between the spectrum of $L_k\left(G\right)$ and that of $L_{k-1}\left(G\right)$. In particular, we show that, for $1\le k\le \lfloor n/2\rfloor$, any eigenvalue λ of $L_k\left(G\right)$ that is not contained in the spectrum of $L_{k-1}\left(G\right)$ satisfies$k\left({\lambda }_2\right(L\left(G\right))-k+1)\le \lambda \le k{\lambda }_n\left(L\right(G\left)\right),$ where ${\lambda }_2\left(L\right(G\left)\right)$ is the second smallest eigenvalue of $L\left(G\right)$ (also known as the algebraic connectivity of G), and ${\lambda }_n\left(L\right(G\left)\right)$ is its largest eigenvalue. Our proof relies on an adaptation of Garland's method, originally developed for the study of high-dimensional Laplacians of simplicial complexes.