  In this paper we generalize the notion of a $k$-graph into (countable)
infinite rank. We then define our $C^*$-algebra in a similar way as in
$k$-graph $C^*$-algebras. With this construction we are able to find analogues
to the Gauge Invariant Uniqueness and Cuntz-Krieger Uniqueness Theorems. We
also show that the $\mathbb{N}$-graph $C^*$-algebras can be viewed as the
inductive limit of $k$-graph $C^*$-algebras. This gives a nice way to describe
the gauge-invariant ideal structure. Additionally, we describe the vertex-set
for regular gauge-invariant ideals of our $N$-graph $C^*$-algebras. We then
take our construction of the $\mathbb{N}$-graph into the algebraic setting and
receive many similarities to the $C^*$-algebra construction.
