Some bounds on the Laplacian eigenvalues of token graphs

The k-token graph F-k (G) of a graph G on n vertices is the graph whose vertices are the ((n)(k)) 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 F-k (G) equals the algebraic connectivity alpha(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 lambda is an eigenvalue of F-k (G), but not of G, then lambda >= k alpha (G)- k + 1. As a consequence, we conclude that if alpha (G) >= k, then alpha(F-h (G)) = alpha(G) for every h <= k. (c) 2024 Published by Elsevier B.V.