  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 allow
correspondences with non-compact left action. In particular, this allows us to
compute the graded $KK$-theory of row-infinite graphs. We develop the theory
necessary 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.
