A general method to find the spectrum and eigenspaces of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e2804" altimg="si219.svg"><mml:mi>k</mml:mi></mml:math>-token graph of a cycle, and 2-token through continuou

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.