Spectral properties of disordered fully-connected graphs

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.