  Motivated by the appearance of fractals in several areas of physics,
especially in solid state physics and the physics of aperiodic order, and in
other sciences, including the quantum information theory, we present a detailed
spectral analysis for a new class of fractal-type diamond graphs, referred to
as bubble-diamond graphs, and provide a gap-labeling theorem in the sense of
Bellissard for the corresponding probabilistic graph Laplacians using the
technique of spectral decimation. Labeling the gaps in the Cantor set by the
normalized eigenvalue counting function, also known as the integrated density
of states, we describe the gap labels as orbits of a second dynamical system
that reflects the branching parameter of the bubble construction and the
decimation structure. The spectrum of the natural Laplacian on limit graphs is
shown generically to be pure point supported on a Cantor set, though one
particular graph has a mixture of pure point and singularly continuous
components.
