### Abstract

This paper considers instability of graphs in all of its possible forms. First, four theorems (one with two interesting special cases) are presented, each of which shows that graphs satisfying certain conditions are unstable. Several infinite families of graphs are investigated. For each family, the four general theorems are used to find and prove enough theorems particular for the family to explain the instability of all graphs in the family up to a certain point. A very dense family of edge-transitive unstable graphs is constructed and, at the other extreme, an unstable graph is constructed whose only symmetry is trivial. Finally, the four theorems are shown to be able to explain all instability.

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

Pages (from-to) | 359-383 |

Number of pages | 25 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 98 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2008 |

### Fingerprint

### Keywords

- Anti-symmetry
- Automorphism group
- Cross-cover
- Graph
- Symmetry
- Unstable graph

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**Unexpected symmetries in unstable graphs.** / Wilson, Stephen E.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 98, no. 2, pp. 359-383. https://doi.org/10.1016/j.jctb.2007.08.001

}

TY - JOUR

T1 - Unexpected symmetries in unstable graphs

AU - Wilson, Stephen E

PY - 2008/3

Y1 - 2008/3

N2 - This paper considers instability of graphs in all of its possible forms. First, four theorems (one with two interesting special cases) are presented, each of which shows that graphs satisfying certain conditions are unstable. Several infinite families of graphs are investigated. For each family, the four general theorems are used to find and prove enough theorems particular for the family to explain the instability of all graphs in the family up to a certain point. A very dense family of edge-transitive unstable graphs is constructed and, at the other extreme, an unstable graph is constructed whose only symmetry is trivial. Finally, the four theorems are shown to be able to explain all instability.

AB - This paper considers instability of graphs in all of its possible forms. First, four theorems (one with two interesting special cases) are presented, each of which shows that graphs satisfying certain conditions are unstable. Several infinite families of graphs are investigated. For each family, the four general theorems are used to find and prove enough theorems particular for the family to explain the instability of all graphs in the family up to a certain point. A very dense family of edge-transitive unstable graphs is constructed and, at the other extreme, an unstable graph is constructed whose only symmetry is trivial. Finally, the four theorems are shown to be able to explain all instability.

KW - Anti-symmetry

KW - Automorphism group

KW - Cross-cover

KW - Graph

KW - Symmetry

KW - Unstable graph

UR - http://www.scopus.com/inward/record.url?scp=38949206508&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38949206508&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2007.08.001

DO - 10.1016/j.jctb.2007.08.001

M3 - Article

VL - 98

SP - 359

EP - 383

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -