Spectral properties of disordered fully-connected graphs

TL;DR

This paper investigates the spectral properties of disordered fully-connected graphs, deriving analytical expressions for spectral density and applying them to electronic and vibrational problems, including bounds for critical parameters.

Contribution

It provides a novel analytical approach to spectral density and eigenvalue estimation for disordered fully-connected graphs, linking to contact process and stochastic diffusion.

Findings

Derived approximate spectral density expressions.

Analyzed eigenvalues for electronic and vibrational problems.

Estimated critical parameters for contact process.

Abstract

The spectral properties of disordered fully-connected graphs with a special type of the node-node interactions are investigated. The approximate analytical expression for the ensemble-averaged spectral density for the Hamiltonian defined on the fully-connected graph is derived and analysed both for the electronic and vibrational problems which can be related to the contact process and to the problem of stochastic diffusion, respectively. It is demonstrated how to evaluate the extreme eigenvalues and use them for finding the lower bound estimates of the critical parameter for the contact process on the disordered fully-connected graphs.