Some Results on the Laplacian Spectra of Token Graphs

Abstract

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 ≤ h ≤ k ≤ 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 G. This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs.