  In this paper we study the spectral features, on fractal-like graphs, of
Hamiltonians which exhibit the special property of perfect quantum state
transfer: the transmission of quantum states without dissipation. The essential
goal is to develop the theoretical framework for understanding the interplay
between perfect quantum state transfer, spectral properties, and the geometry
of the underlying graph, in order to design novel protocols for applications in
quantum information science. We present a new lifting and gluing construction,
and use this to prove results concerning an inductive spectral structure,
applicable to a wide variety of fractal-like graphs. We illustrate this
construction with explicit examples for several classes of diamond graphs.
