On *-representations of a certain class of algebras related to a graph

We study families of self-adjoint operators with given spectra whose sum is a scalar operator. Such families are $*$-representations of certain algebras which can be described in terms of graphs and positive functions on them. The main result is that in the cases where the graph is one of the extended Dynkin graphs $\tilde D_4$, $\tilde E_6$, $\tilde E_7$ or $\tilde E_8$, all irreducible $*$-representations of the corresponding algebra are finite-dimensional. To prove this fact, we introduce the notion of invariant functional on a graph and give their description.