2 research outputs found
Infinite Hamiltonian paths in Cayley diagraphs of hyperbolic symmetry groups
AbstractThe hyperbolic symmetry groups [p,q], [p,q]+, and [p+, q] have certain natural generating sets. We determine whether or not the corresponding Cayley digraphs have one-way infinite or two-way infinite directed Hamiltonian paths. In addition, the analogous Cayley graphs are shown to have both one-way infinite and two-way infinite Hamiltonian paths
Spectral and Combinatorial Aspects of Cayley-Crystals
Owing to their interesting spectral properties, the synthetic crystals over
lattices other than regular Euclidean lattices, such as hyperbolic and fractal
ones, have attracted renewed attention, especially from materials and
meta-materials research communities. They can be studied under the umbrella of
quantum dynamics over Cayley graphs of finitely generated groups. In this work,
we investigate numerical aspects related to the quantum dynamics over such
Cayley graphs. Using an algebraic formulation of the "periodic boundary
condition" due to Lueck [Geom. Funct. Anal. 4, 455-481 (1994)], we devise a
practical and converging numerical method that resolves the true bulk spectrum
of the Hamiltonians. Exact results on the matrix elements of the resolvent,
derived from the combinatorics of the Cayley graphs, give us the means to
validate our algorithms and also to obtain new combinatorial statements. Our
results open the systematic research of quantum dynamics over Cayley graphs of
a very large family of finitely generated groups, which includes the free and
Fuchsian groups.Comment: converging periodic bc for hyperbolic and fractal crystals, tested
against exact result