  In this paper we define the notion of monic representation for the
$C^*$-algebras of finite higher-rank graphs with no sources, and undertake a
comprehensive study of them. Monic representations are the representations
that, when restricted to the commutative $C^*$-algebra of the continuous
functions on the infinite path space, admit a cyclic vector. We link monic
representations to the $\Lambda$-semibranching representations previously
studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal
representation model for nonnegative monic representations.
