A characterization is given for directed graphs that yield graph $C^*$-algebras with continuous trace. This is established for row-finite graphs with no sources first using a groupoid approach, and extended to the general case via the Drinen-Tomforde desingularization. A characterization of continuous-trace AF $C^*$-algebras is obtained. Partial results are given to characterize higher-rank graphs that yield $C^*$-algebras with continuous trace.
"Continuous-trace $k$-graph $C^*$-algebras." Rocky Mountain J. Math. 48 (5) 1511 - 1535, 2018. https://doi.org/10.1216/RMJ-2018-48-5-1511