On the spectra and spectral radii of token graphs

Abstract

AbstractLet G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by $$F_k(G)$$ F k ( G ) , has as vertices the $${n\atopwithdelims ()k}$$ n k k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, $$F_k(K_n)$$ F k ( K n ) is the Johnson graph J(n, k), which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum of $$F_k(G)$$ F k ( G ) in terms of the spectrum of G. For instance, when G is walk-regular, an exact value for the spectral radius $$\rho $$ ρ (or maximum eigenvalue) of $$F_k(G)$$ F k ( G ) is obtained. When G is distance-regular, other eigenvalues of its 2-token graph are derived using the theory of equitable partitions. A generalization of Aldous’ spectral gap conjecture (which is now a theorem) is proposed.