  Let $Q$ be a finite quiver of Dynkin type and $\Lambda=\Lambda_Q$ be the
preprojective algebra of $Q$ over an algebraically closed field $k$. Let
$\mathcal {T}_\Lambda$ be the mutation graph of maximal rigid $\Lambda$
modules. Geiss, Leclerc and Schr$\ddot{\rm o}$er conjectured that $\mathcal
{T}_\Lambda$ is connected, see [C.Geiss, B.Leclerc, J.Schr\"{o}er, Rigid
modules over preprojective algebras, Invent.Math., 165(2006), 589-632]. In this
paper, we prove that this conjecture is true when $\Lambda$ is of
representation finite type or tame type. Moreover, we also prove that $\mathcal
{T}_\Lambda$ is isomorphic to the tilting graph of ${\rm End}_\Lambda T$ for
each maximal rigid $\Lambda$-module $T$ if $\Lambda$ is representation-finite.
