Mutation graphs of maximal rigid modules over finite dimensional preprojective algebras

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.