Sunflower hard disk graphs. (English) Zbl 1527.82003
Summary: The random geometric graph consists of a random point set with links between points with mutual distance below a fixed threshold. Here, we use the same geometric connection rule (“hard disk graph”) but for a deterministic point set, the sunflower spiral. At large distances, the local structure is asymptotically a lattice where for each lattice vector, there is another of length a factor \(\sqrt{5}\) greater, and the angle between these varies log-periodically with distance from the origin. Graph properties including node degrees, stretch factor, clique and chromatic numbers are considered, as well as link formation, connectivity and planarity transitions. Properties depend on a combination of the central region and the perturbed distant lattices, in a rich and varied manner.
MSC:
| 82B10 | Quantum equilibrium statistical mechanics (general) |
| 05C80 | Random graphs (graph-theoretic aspects) |
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