Special characters on star graphs and representations of $*$-algebras
  For a star-shaped graph, we introduce special characters and study their properties. We decompose special characters into odd and even parts and study their evolution under reflections. We apply the obtained formulas to prove that the corresponding $*$-algebra have irreducible infinite-dimensional $*$-representations, if the graph contains an extended Dynkin graph as a proper subgraph. 
Vasyl Ostrovskyi