The paper [PDF] on the upper bound is updated a bit. It will also be updated on the ArXiv. In the video below, there is an update. Both for quivers as well as non-magnetic quivers (quivers without multiple connections), we can use induction to prove results. One of the amendments from the last week was to point out that the lower bound needs the assumption of the graph having no multiple connections. The second amendment was pointing out that for the upper bound valid for Schroedinger operators we need the potential to be non-negative. The talk then discusses two elementary eigenvalue perturbation results. This is in the Schroedinger picture best expressed that the eigenvalues depend in a monotone way both on the scalar potential (modeled by loops) and vector potential (modeled by multiple connections in the graph).