On maximal gradings of simply connected algebras (Q801006)

In [Invent. Math. 65, 331-378 (1982; Zbl 0482.16026)] \textit{K. Bongartz} and \textit{P. Gabriel} proved that there is a bijection between the isomorphism classes of (representation-finite) simply connected algebras and the isomorphism classes of representation-finite graded trees. Denote by G(n) the maximal value of gradings through all representation-finite graded trees with n vertices and by F(n) the maximal length of the Auslander-Reiten quivers of the associated simply connected algebras. In the paper an estimation of the numbers G(n) and F(n) is given. Moreover an upper bound on the number of nonisomorphic indecomposable modules over simply connected basic algebras with n simple modules is obtained.