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 scarlike 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 nonperturbative 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 nonperturbative and mid-spectrum bands, due to the scars, present in the latter. The applications of our findings to $2\mathrm{d}$ quantum gravity are discussed.

• Received 30 May 2023
• Revised 15 August 2023
• Accepted 15 August 2023

DOI:https://doi.org/10.1103/PhysRevB.108.094203