Some bounds on the Laplacian eigenvalues of token graphs

Abstract

The k-token graph $F_k\left(G\right)$ of a graph G on n vertices is the graph whose vertices are the $\left(\begin{array}{c}n\\ k\end{array}\right)$ k-subsets of vertices from G, two of which are 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 $F_k\left(G\right)$ equals the algebraic connectivity $\alpha \left(G\right)$ of G.

In this paper, we give some bounds on the (Laplacian) eigenvalues of the k-token graph (including the algebraic connectivity) in terms of the h-token graph, with $h\le k$. For instance, we prove that if λ is an eigenvalue of $F_k\left(G\right)$, but not of G, then$\lambda \ge k\alpha \left(G\right)-k+1.$ As a consequence, we conclude that if $\alpha \left(G\right)\ge k$, then $\alpha \left(F_h\right(G\left)\right)=\alpha \left(G\right)$ for every $h\le k$.