  We consider the properties of the random regular graph with node degree $d$
perturbed by chemical potentials $\mu_k$ for a number of short $k$-cycles. We
analyze both numerically and analytically the phase diagram of the model in the
$(\mu_k,d)$ plane. The critical curve separating the homogeneous and
clusterized phases is found and it is demonstrated that the clusterized phase
itself generically is separated as the function of $d$ into the phase with
ideal clusters and phase with coupled ones when the continuous spectrum gets
formed. The eigenstate spatial structure of the model is investigated and it is
found that there are localized scar-like states in the delocalized part of the
spectrum, that are related to the topologically equivalent nodes in the graph.
We also reconsider the localization of the states in the non-perturbative band
formed by eigenvalue instantons and find the semi-Poisson level spacing
distribution. The Anderson transition for the case of combined ($k$-cycle)
structural and diagonal (Anderson) disorders is investigated. It is found that
the critical diagonal disorder gets reduced sharply at the clusterization phase
transition, but does it unevenly in non-perturbative and mid-spectrum bands,
due to the scars, present in the latter. The applications of our findings to
$2$d quantum gravity are discussed.
