  We study the Laplacian operator of an uncorrelated random network and, as an
application, consider hopping processes (diffusion, random walks, signal
propagation, etc.) on networks. We develop a strict approach to these problems.
We derive an exact closed set of integral equations, which provide the averages
of the Laplacian operator's resolvent. This enables us to describe the
propagation of a signal and random walks on the network. We show that the
determining parameter in this problem is the minimum degree $q_m$ of vertices
in the network and that the high-degree part of the degree distribution is not
that essential. The position of the lower edge of the Laplacian spectrum
$\lambda_c$ appears to be the same as in the regular Bethe lattice with the
coordination number $q_m$. Namely, $\lambda_c>0$ if $q_m>2$, and $\lambda_c=0$
if $q_m\leq2$. In both these cases the density of eigenvalues
$\rho(\lambda)\to0$ as $\lambda\to\lambda_c+0$, but the limiting behaviors near
$\lambda_c$ are very different. In terms of a distance from a starting vertex,
the hopping propagator is a steady moving Gaussian, broadening with time. This
picture qualitatively coincides with that for a regular Bethe lattice. Our
analytical results include the spectral density $\rho(\lambda)$ near
$\lambda_c$ and the long-time asymptotics of the autocorrelator and the
propagator.
