Some bounds on the Laplacian eigenvalues of token graphs

TL;DR

This paper establishes bounds on the Laplacian eigenvalues of k-token graphs, relating them to the eigenvalues of smaller token graphs and providing conditions under which the algebraic connectivity remains unchanged.

Contribution

It introduces new bounds on Laplacian eigenvalues of k-token graphs based on smaller token graphs, extending understanding of their spectral properties.

Findings

Eigenvalues of F_k(G) are bounded in terms of the algebraic connectivity of G.

If α(G) ≥ k, then the algebraic connectivity of F_h(G) equals that of G for all h ≤ k.

Eigenvalues not of G satisfy λ ≥ kα(G) - k + 1.

Abstract

The $k$-token graph $F_k(G)$ of a graph $G$ on $n$ vertices is the graph whose vertices are the ${n\choose k}$ $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. It is known that the algebraic connectivity (or second Laplacian eigenvalue) of $F_k(G)$ equals the algebraic connectivity $\alpha(G)$ of $G$. In this paper, we give some bounds on the (Laplacian) eigenvalues of a $k$-token graph (including the algebraic connectivity) in terms of the $h$-token graph, with $h\leq k$. For instance, we prove that if $\lambda$ is an eigenvalue of $F_k(G)$, but not of $G$, then $$ \lambda\ge k\alpha(G)-k+1. $$ As a consequence, we conclude that if $\alpha(G)\geq k$, then $\alpha(F_h(G))=\alpha(G)$ for every $h\le k$.