  In this article we provide an identification between the wavelet
decompositions of certain fractal representations of $C^*-$algebras of directed
graphs of M. Marcolli and A. Paolucci, and the eigenspaces of Laplacians
associated to spectral triples constructed from Cantor fractal sets that are
the infinite path spaces of Bratteli diagrams associated to the
representations, with a particular emphasis on wavelets for representations of
$\mathcal{O}_D$. In particular, in this setting we use results of J. Pearson
and J. Bellissard, and A. Julien and J. Savinien, to construct first the
spectral triple and then the Laplace Beltrami operator on the associated Cantor
set. We then prove that in certain cases, the orthogonal wavelet decomposition
and the decomposition via orthogonal eigenspaces match up precisely. We give
several explicit examples, including an example related to a Sierpinski
fractal, and compute in detail all the eigenvalues and corresponding
eigenspaces of the Laplace Beltrami operators for the equal weight case for
representations of Cuntz algebras, and in the uneven weight case for certain
representations of $\mathcal{O}_2$, and show how the eigenspaces and wavelet
subspaces at different levels are related.
