2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/144418Given a row-finite $k$-graph $Λ$ with no sources we investigate the $K$-theory of the higher rank graph $C^*$-algebra, $C^*(Λ)$. When $k=2$ we are able to give explicit formulae to calculate the $K$-groups of $C^*(Λ)$. The $K$-groups of $C^*(Λ)$ for $k>2$ can be calculated under certain circumstances and we consider the case $k=3$. We prove that for arbitrary $k$, the torsion-free rank of $K_0(C^*(Λ))$ and $K_1(C^*Λ))$ are equal when $C^*(Λ)$ is unital, and for $k=2$ we determine the position of the class of the unit of $C^*(Λ)$ in $K_0(C^*(Λ))$.23 pages. To appear in the New York Journal of Mathematics (http://nyjm.albany.edu:8000/). Revisions include: a different numbering system for sections, theorems and related parts; correction of typographical errors; re-organisation of results; and addition of examples (Section 5)Operator AlgebrasK-Theory and Homology46L80 (Primary); 46L35 (Secondary)text