Graded $K$-theory and $K$-homology of relative Cuntz-Pimsner algebras and graph $C^*$-algebras

Abstract

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.