On the spectra and spectral radii of token graphs

TL;DR

This paper investigates the spectral properties of token graphs derived from a base graph G, providing formulas for eigenvalues and spectral radius, especially for walk-regular and distance-regular graphs, and proposes a generalization of Aldous' spectral gap conjecture.

Contribution

It offers new spectral formulas for token graphs based on the spectrum of G, extending understanding of their eigenvalues and spectral gaps.

Findings

Exact spectral radius for walk-regular graphs' token graphs.

Eigenvalues of 2-token graphs for distance-regular graphs.

Proposed a generalization of Aldous' spectral gap conjecture.

Abstract

Let $G$ be a graph on $n$ vertices. The $k$-token graph (or symmetric $k$-th power) of $G$, denoted by $F_k(G)$ has as vertices the ${n\choose k}$ $k$-subsets of vertices from $G$, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in $G$. In particular, $F_k(K_n)$ 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 $F_k(G)$ in terms of the spectrum of $G$. For instance, when $G$ is walk-regular, an exact value for the spectral radius $\rho$ (or maximum eigenvalue) of $F_k(G)$ 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.