Graded K-theory and K-homology of relative Cuntz–Pimsner algebras and graph C^⁎-algebras

We establish exact sequences in KK-theory for graded relative Cuntz–Pimsner algebras associated to nondegenerate C -correspondences. We use this to calculate the graded K-theory and K-homology of relative Cuntz–Krieger algebras of directed graphs for gradings induced by {0,1}–valued labellings of their edge sets. ⁎