  We study a canonical C$^*$-algebra, $\mathcal{S}(\Gamma, \mu)$, that arises
from a weighted graph $(\Gamma, \mu)$, specific cases of which were previously
studied in the context of planar algebras. We discuss necessary and sufficient
conditions of the weighting which ensure simplicity and uniqueness of trace of
$\mathcal{S}(\Gamma, \mu)$, and study the structure of its positive cone. We
then study the $*$-algebra, $\mathcal{A}$, generated by the generators of
$\mathcal{S}(\Gamma, \mu)$, and use a free differential calculus and techniques
of Charlesworth and Shlyakhtenko, as well as Mai, Speicher, and Weber to show
that certain "loop" elements have no atoms in their spectral measure. After
modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint
elements $x \in M_{n}(\mathcal{A})$ have algebraic Cauchy transform, we explore
some applications to eigenvalues of polynomials in Wishart matrices and to
diagrammatic elements in von Neumann algebras initially considered by Guionnet,
Jones, and Shlyakhtenko.
