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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. Weanalyze both numerically and analytically the phase diagram of the model in the$(\mu_k,d)$ plane. The critical curve separating the homogeneous andclusterized phases is found and it is demonstrated that the clusterized phaseitself generically is separated as the function of $d$ into the phase withideal clusters and phase with coupled ones when the continuous spectrum getsformed. The eigenstate spatial structure of the model is investigated and it isfound that there are localized scar-like states in the delocalized part of thespectrum, that are related to the topologically equivalent nodes in the graph.We also reconsider the localization of the states in the non-perturbative bandformed by eigenvalue instantons and find the semi-Poisson level spacingdistribution. The Anderson transition for the case of combined ($k$-cycle)structural and diagonal (Anderson) disorders is investigated. It is found thatthe critical diagonal disorder gets reduced sharply at the clusterization phasetransition, 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.

Anatomy of the fragmented Hilbert space: eigenvalue tunneling, quantum scars and localization in the perturbed random regular graph