  Here, we focus on Anderson type operators over infinite graphs where the
randomness acts through higher rank perturbations. We show that for special
family of graphs, the operator has non-trivial multiplicity for its pure point
spectrum. We, also, show that for some family of graphs, any unitary which
fixes the random operator, arising from an automorphism of the graph is
identity; but that, for these graphs the spectrum of the random operator has
non-trivial multiplicity.
