The spectral problem and algebras associated with extended Dynkin   graphs. I
  There is a connection between *-representations of algebras associated with graphs and the problem about the spectrum of a sum of Hermitian operators (spectral problem). For algebras associated with extended Dynkin graphs we give an explicit description of the parameters for which there are $*$-representations and an algorithm for constructing these representations. 
Stanislav Krugljak, Stanislav Popovych, Yurii Samoilenko