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In this work we analytically explain the origin of the mobility edge in theensemble of random regular graphs (RRG), with the connectivity $d$ and thefraction $\beta$ of disordered nodes, the location of which is under control.It is shown that the mobility edge in the spectrum survives in a certain rangeof parameters $(d,\beta)$ at infinitely large uniformly distributed disorder.The critical curve separating extended and localized states is derivedanalytically and confirmed numerically. The duality in the localizationproperties between the sparse and extremely dense RRG has been found andunderstood. The mobility edge physics has been analyzed numerically for theabove partially disordered RRG, perturbed by the non-reciprocity parameter ofnode as well as by the enhanced number of short cycles, usually almost absenton RRG.

Robust extended states in Anderson model on partially disordered random regular graphs