The exact Laplacian spectrum for the Dyson hierarchical network

TL;DR

This paper derives the exact eigenvalues and eigenvectors of the Laplacian matrix for the Dyson hierarchical graph, enabling analytical study of dynamic processes like random walks and vibrational modes on this complex network.

Contribution

The authors explicitly compute the entire spectrum of the Laplacian for the Dyson hierarchical graph, a significant advancement for analyzing dynamics on such networks.

Findings

Eigenvalues and eigenvectors of the Laplacian are explicitly derived.

Analytical expressions for random walk and quantum walk behaviors.

Insights into relaxation times of polymer structures modeled by the graph.

Abstract

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 $\mathcal{G}$.