### Abstract

Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with other graphs called replacement graphs to obtain a new graph R ( G ). By iterating this procedure, a sequence of graphs { R^{n} ( G ) } is obtained. When each graph in this sequence is normalized to have diameter one, the resulting sequence may converge in the Gromov-Hausdorff metric. In this paper, we compute the topological dimension of limit spaces of normalized sequences of iterated vertex replacements involving more than one replacement graph. We also give examples of vertex replacement rules that yield fractals.

Original language | English (US) |
---|---|

Pages (from-to) | 2013-2025 |

Number of pages | 13 |

Journal | Topology and its Applications |

Volume | 153 |

Issue number | 12 |

DOIs | |

State | Published - Jun 1 2006 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*153*(12), 2013-2025. https://doi.org/10.1016/j.topol.2005.07.008

}

*Topology and its Applications*, vol. 153, no. 12, pp. 2013-2025. https://doi.org/10.1016/j.topol.2005.07.008

**The topological dimension of limits of vertex replacements.** / Previte, Michelle; Yang, Shun Hsiang.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The topological dimension of limits of vertex replacements

AU - Previte, Michelle

AU - Yang, Shun Hsiang

PY - 2006/6/1

Y1 - 2006/6/1

N2 - Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with other graphs called replacement graphs to obtain a new graph R ( G ). By iterating this procedure, a sequence of graphs { Rn ( G ) } is obtained. When each graph in this sequence is normalized to have diameter one, the resulting sequence may converge in the Gromov-Hausdorff metric. In this paper, we compute the topological dimension of limit spaces of normalized sequences of iterated vertex replacements involving more than one replacement graph. We also give examples of vertex replacement rules that yield fractals.

AB - Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with other graphs called replacement graphs to obtain a new graph R ( G ). By iterating this procedure, a sequence of graphs { Rn ( G ) } is obtained. When each graph in this sequence is normalized to have diameter one, the resulting sequence may converge in the Gromov-Hausdorff metric. In this paper, we compute the topological dimension of limit spaces of normalized sequences of iterated vertex replacements involving more than one replacement graph. We also give examples of vertex replacement rules that yield fractals.

UR - http://www.scopus.com/inward/record.url?scp=33646787502&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646787502&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2005.07.008

DO - 10.1016/j.topol.2005.07.008

M3 - Article

VL - 153

SP - 2013

EP - 2025

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 12

ER -