On functions on graphs and representations of a certain class of ∗-algebras

We study ∗-representations of certain algebras which can be described in terms of graphs and positive functions (weights) on the set of their vertices, continuing an earlier investigation of the case where the graph is a Dynkin graph or an extended Dynkin graph with a weight of a special kind.For the cases where the graph is one of the extended Dynkin graphs , , or , we prove that all irreducible ∗-representations of the corresponding algebras are finite-dimensional.In the case of a graph which properly contains an extended Dynkin graph, we study the evolution of weights under the action of the Coxeter functors, in particular, we show that there exist two linearly independent p-invariant weights. We also prove that there exists a weight which makes the corresponding algebra to have an infinite-dimensional irreducible ∗-representation.