  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.
