  We study the algebraic connectivity (or second Laplacian eigenvalue) of token
graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ 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 $G$. Recently, it was conjectured that the algebraic
connectivity of $F_k(G)$ equals the algebraic connectivity of $G$. In this
paper, we prove the conjecture for new infinite families of graphs, such as
trees and graphs with maximum degree large enough.
