Growth of selfsimilar graphs
Abstract
Locally finite selfsimilar graphs with bounded geometry and without bounded geometry as well as nonlocally finite selfsimilar 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 $\nu$ and the volume scaling factor $\mu$ can be defined similarly to the corresponding parameters of continuous selfsimilar sets. There are different notions of growth dimensions of graphs. For a rather general class of selfsimilar 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 selfsimilar fractals: \[\dim X=\frac{\log \mu}{\log \nu}.\]
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2002
 DOI:
 10.48550/arXiv.math/0202171
 arXiv:
 arXiv:math/0202171
 Bibcode:
 2002math......2171K
 Keywords:

 Combinatorics;
 05C12;
 28A80
 EPrint:
 14 pages, 3 figures