On the spectra of token graphs of cycles and other 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 $G$. 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 $F_2(\textit{}C_n)$.