  In this paper, we present a new way to associate a finitely summable spectral
triple to a higher-rank graph $\Lambda$, via the infinite path space
$\Lambda^\infty$ of $\Lambda$. Moreover, we prove that this spectral triple has
a close connection to the wavelet decomposition of $\Lambda^\infty$ which was
introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the
concept of stationary $k$-Bratteli diagrams, in order to associate a family of
ultrametric Cantor sets, and their associated Pearson-Bellissard spectral
triples, to a finite, strongly connected higher-rank graph $\Lambda$. We then
study the zeta function, abscissa of convergence, and Dixmier trace associated
to the Pearson-Bellissard spectral triples of these Cantor sets, and show these
spectral triples are $\zeta$-regular in the sense of Pearson and Bellissard. We
obtain an integral formula for the Dixmier trace given by integration against a
measure $\mu$, and show that $\mu$ is a rescaled version of the measure $M$ on
$\Lambda^\infty$ which was introduced by an Huef, Laca, Raeburn, and Sims.
Finally, we investigate the eigenspaces of a family of Laplace-Beltrami
operators associated to the Dirichlet forms of the spectral triples. We show
that these eigenspaces refine the wavelet decomposition of $L^2(\Lambda^\infty,
M)$ which was constructed by Farsi et al.
