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,…, n r for all n 1, n 2,…, 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 (C n).