The exact Laplacian spectrum for the Dyson hierarchical network

We consider the Dyson hierarchical graph [Image: see text], that is a weighted fully-connected graph, where the pattern of weights is ruled by the parameter σ ∈ (1/2, 1]. Exploiting the deterministic recursivity through which [Image: see text] 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 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 [Image: see text], the relaxation times of a polymer whose structure is described by [Image: see text], and the community structure of [Image: see text] in terms of modularity measures.