2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/170129We summarize recent works on the stability under disorder of the absolutely continuous spectra of random operators on tree graphs. The cases covered include: Schrödinger operators with random potential, quantum graph operators for trees with randomized edge lengths, and radial quasi-periodic operators perturbed by random potentials.The article covers talks given at UAB Int. Conf. on Diff. Eq. and Math. Phys., Birmingham, March 2005 (RS and SW), and at AMS Snowbird Conference, Utah, June 2005 (MA)Mathematical Physics82B44; 47B80text