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In this thesis we generalise the six-term exact sequence in graded$KK$-theory obtained in a paper of Kumjian, Pask and Sims (2017) to allowcorrespondences with non-compact left action. In particular, this allows us tocompute the graded $KK$-theory of row-infinite graphs. We develop the theorynecessary for following the arguments of Kumjian, Pask and Sims and of Pimsner(1997), with detailed sections on Hilbert modules, $C^*$-correspondences,Crossed products, Toeplitz algebras, Cuntz-Pimsner algebras and $KK$-theory.

Honours thesis: Exact sequences in graded $KK$-theory for Cuntz-Pimsner algebras