On the spectra of token graphs of cycles and other graphs - Reyes, M.A.; Dalfó, C.; Fiol, M.A.; Messegué, A. (Linear algebra and its applications, 2023)
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. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of Fk(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 Or for all r, and the multipartite complete graphs Kn1,n2,…,nr for all n1,n2,…,nr In the case of cycles, we present a new method that allows us to compute the whole spectrum of F2(Cn). This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of F2(Cn).