  A graded quiver with superpotential is a quiver whose arrows are assigned
degrees $c\in \{0, 1, \cdots, m\}$, for some integer $m \geq 0$, with relations
generated by a superpotential of degree $m-1$. Ordinary quivers ($m=1)$ often
describe the open string sector of D-brane systems; in particular, they capture
the physics of D3-branes at local Calabi-Yau (CY) 3-fold singularities in type
IIB string theory, in the guise of 4d $\mathcal{N}=1$ supersymmetric quiver
gauge theories. It was pointed out recently that graded quivers with $m=2$ and
$m=3$ similarly describe systems of D-branes at CY 4-fold and 5-fold
singularities, as 2d $\mathcal{N}=(0,2)$ and 0d $\mathcal{N}=1$ gauge theories,
respectively. In this work, we further explore the correspondence between
$m$-graded quivers with superpotential, $Q_{(m)}$, and CY $(m+2)$-fold
singularities, ${\mathbf X}_{m+2}$. For any $m$, the open string sector of the
topological B-model on ${\mathbf X}_{m+2}$ can be described in terms of a
graded quiver. We illustrate this correspondence explicitly with a few infinite
families of toric singularities indexed by $m \in \mathbb{N}$, for which we
derive "toric" graded quivers associated to the geometry, using several
complementary perspectives. Many interesting aspects of supersymmetric quiver
gauge theories can be formally extended to any $m$; for instance, for one
family of singularities, dubbed $C(Y^{1,0}(\mathbb{P}^m))$, that generalizes
the conifold singularity to $m>1$, we point out the existence of a formal
"duality cascade" for the corresponding graded quivers.
