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 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. In this paper, 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≤n2, the Laplacian spectrum of Fh(G) is contained in the Laplacian spectrum of Fk(G). We also show that the doubled odd graphs and doubled Johnson graphs can be obtained as token graphs of the complete graph Kn and the star Sn=K1,n−1, respectively. 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 to tokens graphs a well-known property stating that the Laplacian eigenvalues of G are closely related to the Laplacian eigenvalues of G‾. Finally, the doubled odd graphs and doubled Johnson graphs provide two infinite families, together with some others, in which the algebraic connectivities of the original graph and its token graph coincide. Moreover, we conjecture that this is the case for any graph G and its token graph.