  We consider the Dyson hierarchical graph $\mathcal{G}$, that is a weighted
fully-connected graph, where the pattern of weights is ruled by the parameter
$\sigma \in (1/2, 1]$. Exploiting the deterministic recursivity through which
$\mathcal{G}$ is built, we are able to derive explicitly the whole set of the
eigenvalues and the eigenvectors for its Laplacian matrix. Given that the
Laplacian operator is intrinsically implied in the analysis of dynamic
processes (e.g., random walks) occurring on the graph, as well as in the
investigation of the dynamical properties of connected structures themselves
(e.g., vibrational structures and the relaxation modes), this result allows
addressing analytically a large class of problems. In particular, as examples
of applications, we study the random walk and the continuous-time quantum walk
embedded in $\mathcal{G}$, and the relaxation times of a polymer whose
structure is described by $\mathcal{G}$.
