Growth of Self-Similar Graphs

✍ Scribed by B. Krön

• Publisher: John Wiley and Sons
• Year: 2004
• Tongue: English
• Weight: 134 KB
• Volume: 45
• Category: Article
• ISSN: 0364-9024
• DOI: 10.1002/jgt.10157

✦ Synopsis

Abstract

Locally finite self‐similar graphs with bounded geometry and without bounded geometry as well as non‐locally finite self‐similar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The length scaling factor ν and the volume scaling factor μ can be defined similarly to the corresponding parameters of continuous self‐similar sets. There are different notions of growth dimensions of graphs. For a rather general class of self‐similar graphs, it is proved that all these dimensions coincide and that they can be calculated in the same way as the Hausdorff dimension of continuous self‐similar fractals: ${\rm dim}\ X={{\rm log}\ \mu \over {\rm log}\ \nu }$. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 224–239, 2004