  We study the properties of random graphs where for each vertex a {\it
neighbourhood} has been previously defined. The probability of an edge joining
two vertices depends on whether the vertices are neighbours or not, as happens
in Small World Graphs (SWGs). But we consider the case where the average degree
of each node is of order of the size of the graph (unlike SWGs, which are
sparse). This allows us to calculate the mean distance and clustering, that are
qualitatively similar (although not in such a dramatic scale range) to the case
of SWGs. We also obtain analytically the distribution of eigenvalues of the
corresponding adjacency matrices. This distribution is discrete for large
eigenvalues and continuous for small eigenvalues. The continuous part of the
distribution follows a semicircle law, whose width is proportional to the
"disorder" of the graph, whereas the discrete part is simply a rescaling of the
spectrum of the substrate. We apply our results to the calculation of the
mixing rate and the synchronizability threshold.
