The spectral problem and *-representations of algebras associated with   Dynkin graphs
  We study the connection between *-representations of algebras associated with graphs, locally-scalar graph representations and the problem about the spectrum of a sum of two Hermitian operators. For algebras associated with Dynkin graphs we give an explicit description of the parameters for which there are irreducible representations and an algorithm for contructing these representations. 
Stanislav Krugljak, Stanislav Popovych, Yurii Samoilenko