Some Results on the Laplacian Spectra of Token Graphs

We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The k-token graph \(F_k(G)\) 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 work, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers h and k such that \(1\le h\le k\le \frac{n}{2}\), the Laplacian spectrum of \(F_h(G)\) is contained in the Laplacian spectrum of \(F_k(G)\). Besides, we obtain a relationship between the spectra of the k-token graph of G and the k-token graph of its complement \(\overline{G}\). This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs.