  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 $G$. It is a known
result that the algebraic connectivity (or second Laplacian eigenvalue) of
$F_k(G)$ equals the algebraic connectivity of $G$.
  In this paper, we first give results that relate the algebraic connectivities
of a token graph and the same graph after removing a vertex. Then, we prove the
result on the algebraic connectivity of 2-token graphs for two infinite
families: the odd graphs $O_r$ for all $r$, and the multipartite complete
graphs $K_{n_1,n_2,\ldots,n_r}$ for all $n_1,n_2,\ldots,n_r$ In the case of
cycles, we present a new method that allows us to compute the whole spectrum of
$F_2(C_n)$. This method also allows us to obtain closed formulas that give
asymptotically exact approximations for most of the eigenvalues of
$F_2(\textit{}C_n)$.
