  In this paper, we present two new ways to associate a spectral triple to a
higher-rank graph $\Lambda$. Moreover, we prove that these spectral triples are
intimately connected to the wavelet decomposition of the infinite path space of
$\Lambda$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We
first introduce the concept of stationary $k$-Bratteli diagrams, to associate a
family of ultrametric Cantor sets to a finite, strongly connected higher-rank
graph $\Lambda$. Then we show that under mild hypotheses, the
Pearson-Bellissard spectral triples of such Cantor sets have a regular
$\zeta$-function, whose abscissa of convergence agrees with the Hausdorff
dimension of the Cantor set, and that the measure $\mu$ induced by the
associated Dixmier trace agrees with the measure $M$ on the infinite path space
$\Lambda^\infty$ of $\Lambda$ which was introduced by an Huef, Laca, Raeburn,
and Sims. Furthermore, we prove that $\mu = M$ is a rescaled version of the
Hausdorff measure of the ultrametric Cantor set. From work of Julien and
Savinien, we know that for $\zeta$-regular Pearson-Bellissard spectral triples,
the eigenspaces of the associated Laplace-Beltrami operator constitute an
orthogonal decomposition of $L^2(\Lambda^\infty, \mu)$; we show that this
orthogonal decomposition refines the wavelet decomposition of Farsi et al. In
addition, we generalize a spectral triple of Consani and Marcolli from
Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras, and prove that the
wavelet decomposition of Farsi et al.~describes the eigenspaces of its Dirac
operator.
