  We study the Laplacian spectrum of token graphs, also called symmetric powers
of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose
vertices are the $k$-subsets of vertices from $G$, two of which being adjacent
whenever their symmetric difference is a pair of adjacent vertices in $G$. In
this paper, we give a relationship between the Laplacian spectra of any two
token graphs of a given graph. In particular, we show that, for any integers
$h$ and $k$ such that $1\le h\le k\le \frac{n}{2}$, the Laplacian spectrum of
$F_h(G)$ is contained in the Laplacian spectrum of $F_k(G)$. We also show that
the double odd graphs and doubled Johnson graphs can be obtained as token
graphs of the complete graph $K_n$ and the star $S_{n}=K_{1,n-1}$,
respectively. Besides, we obtain a relationship between the spectra of the
$k$-token graph of $G$ and the $k$-token graph of its complement
$\overline{G}$. This generalizes a well-known property for Laplacian
eigenvalues of graphs to token graphs. Finally, the double odd graphs and
doubled Johnson graphs provide two infinite families, together with some
others, in which the algebraic connectivities of the original graph and its
token graph coincide. Moreover, we conjecture that this is the case for any
graph $G$ and its token graph.
