Some bounds on the Laplacian eigenvalues of token graphs - Dalfó, C.; Fiol, M.A.; Messegué, A. (Discrete mathematics, 2025)
The k-token graph Fk(G) of a graph G on n vertices is the graph whose vertices are the (nk)k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is known that the algebraic connectivity (or second Laplacian eigenvalue) of Fk(G) equals the algebraic connectivity α(G) of G. In this paper, we give some bounds on the (Laplacian) eigenvalues of the k-token graph (including the algebraic connectivity) in terms of the h-token graph, with h≤k. For instance, we prove that if λ is an eigenvalue of Fk(G), but not of G, thenλ≥kα(G)−k+1. As a consequence, we conclude that if α(G)≥k, then α(Fh(G))=α(G) for every h≤k.