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In this paper, we discuss a method of constructing separable representationsof the $C^*$-algebras associated to strongly connected row-finite $k$-graphs$\Lambda$. We begin by giving an alternative characterization of the$\Lambda$-semibranching function systems introduced in an earlier paper, withan eye towards constructing such representations that are faithful. Our newcharacterization allows us to more easily check that examples satisfy certainnecessary and sufficient conditions. We present a variety of new examplesrelying on this characterization. We then use some of these methods and adirect limit procedure to construct a faithful separable representation for anyrow-finite source-free $k$-graph.

Representations of higher-rank graph $C^*$-algebras associated to $\Lambda$-semibranching function systems