Equivariant Dimensions of Graph C*-algebras

We explore the recently introduced local-triviality dimensions by studying gauge actions on graph $C^{\ast }$-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For $C^{\ast }$-algebras of finite acyclic graphs and finite cycles, we characterize the finiteness of these dimensions, and we further study the gauge actions on many examples of graph $C^{\ast }$-algebras. These include the Toeplitz algebra, Cuntz algebras, and $q$-deformed spheres.