2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/174652We provide a detailed analysis of atomic *-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal *-dilations of defect free representations modelled on finite groups of rank 2.42 pages, 5 figuresOperator Algebras47L55text