Independence and matching numbers of some token graphs

Let $G$ be a graph of order $n$ and let $k\in\{1,\ldots,n-1\}$. The $k$-token graph $F_k(G)$ of $G$, is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. We study the independence and matching numbers of $F_k(G)$. We present a tight lower bound for the matching number of $F_k(G)$ for the case in which $G$ has either a perfect matching or an almost perfect matching. Also, we estimate the independence number for bipartite $k$-token graphs, and determine the exact value for some graphs.