Mutation graphs of maximal rigid modules over finite dimensional preprojective algebras

Let $Q$ be a finite quiver of Dynkin type and $\mathrm{\Lambda }={\mathrm{\Lambda }}_Q$ be the preprojective algebra of $Q$ over an algebraically closed field $k$. Let ${\mathcal{T}}_{\mathrm{\Lambda }}$ be the mutation graph of maximal rigid $\mathrm{\Lambda }$ modules. Geiss, Leclerc and Schr$\ddot{\mathrm{o}}$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 $\mathrm{\Lambda }$ is of representation finite type or tame type. Moreover, we also prove that $\mathcal {T}\Lambda$isisomorphictothetiltinggraphof${\rm End}_\Lambda T$ for each maximal rigid $\mathrm{\Lambda }$-module $T$ if $\mathrm{\Lambda }$ is representation-finite.