Exactly solvable model with two conductor-insulator transitions driven by impurities

We present an exact analysis of two conductor-insulator transitions in the random graph model where low connectivity means high impurity concentration. The adjacency matrix of the random graph is used as a hopping Hamiltonian. We compute the height of the delta peak at zero energy in its spectrum exactly and describe analytically the structure and contribution of localized eigenvectors. The system is a conductor for average connectivities between 1.421 529... and 3.154 985... but an insulator in the other regimes. We explain the spectral singularity at average connectivity e = 2.718 281... and relate it to other enumerative problems in random graph theory.