  The $k$-token graph $F_k(G)$ of a graph $G$ on $n$ vertices is the graph
whose vertices are the ${n\choose k}$ $k$-subsets of vertices from $G$, two of
which being adjacent whenever their symmetric difference is a pair of adjacent
vertices in $G$.
  It is known that the algebraic connectivity (or second Laplacian eigenvalue)
of $F_k(G)$ equals the algebraic connectivity $\alpha(G)$ of $G$.
  In this paper, we give some bounds on the (Laplacian) eigenvalues of a
$k$-token graph (including the algebraic connectivity) in terms of the
$h$-token graph, with $h\leq k$. For instance, we prove that if $\lambda$ is an
eigenvalue of $F_k(G)$, but not of $G$, then $$ \lambda\ge k\alpha(G)-k+1. $$
As a consequence, we conclude that if $\alpha(G)\geq k$, then
$\alpha(F_h(G))=\alpha(G)$ for every $h\le k$.

Combinatorics