A general method to find the spectrum and eigenspaces of the k-token of a cycle, and 2-token through continuous fractions

The k-token graph Fₖ(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 propose a general method to find the spectrum and eigenspaces of the k-token graph Fₖ(Cₙ) of a cycle Cₙ. The method is based on the theory of lift graphs and the recently introduced theory of over-lifts. In the case of $k=2$, we use continuous fractions to derive the spectrum and eigenspaces of the 2-token graph of Cₙ.