Gaps labeling theorem for the bubble-diamond self-similar graphs

Abstract

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.