### Abstract

Original language | English |
---|---|

Pages (from-to) | 275–291 |

Number of pages | 17 |

Journal | Ars Mathematica Contemporanea |

Volume | 13 |

Issue number | 2 |

Early online date | 9 Mar 2017 |

Publication status | E-pub ahead of print - 9 Mar 2017 |

### Fingerprint

### Keywords

- degree-diameter problem
- sumsets
- circulant graphs
- Cayley graphs

### Cite this

*Ars Mathematica Contemporanea*,

*13*(2), 275–291.

}

*Ars Mathematica Contemporanea*, vol. 13, no. 2, pp. 275–291.

**Large circulant graphs of fixed diameter and arbitrary degree.** / Bevan, David; Erskine, Grahame; Lewis, Robert.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Large circulant graphs of fixed diameter and arbitrary degree

AU - Bevan, David

AU - Erskine, Grahame

AU - Lewis, Robert

PY - 2017/3/9

Y1 - 2017/3/9

N2 - We consider the degree-diameter problem for undirected and directed circulant graphs. To date, attempts to generate families of large circulant graphs of arbitrary degree for a given diameter have concentrated mainly on the diameter 2 case. We present a direct product construction yielding improved bounds for small diameters and introduce a new general technique for “stitching” together circulant graphs which enables us to improve the current best known asymptotic orders for every diameter. As an application, we use our constructions in the directed case to obtain upper bounds on the minimum size of a subset A of a cyclic group of order n such that the k-fold sumset kA is equal to the whole group. We also present a revised table of largest known circulant graphs of small degree and diameter.

AB - We consider the degree-diameter problem for undirected and directed circulant graphs. To date, attempts to generate families of large circulant graphs of arbitrary degree for a given diameter have concentrated mainly on the diameter 2 case. We present a direct product construction yielding improved bounds for small diameters and introduce a new general technique for “stitching” together circulant graphs which enables us to improve the current best known asymptotic orders for every diameter. As an application, we use our constructions in the directed case to obtain upper bounds on the minimum size of a subset A of a cyclic group of order n such that the k-fold sumset kA is equal to the whole group. We also present a revised table of largest known circulant graphs of small degree and diameter.

KW - degree-diameter problem

KW - sumsets

KW - circulant graphs

KW - Cayley graphs

UR - http://amc-journal.eu/index.php/amc/article/view/969

M3 - Article

VL - 13

SP - 275

EP - 291

JO - Ars Mathematica Contemporanea

JF - Ars Mathematica Contemporanea

SN - 1855-3966

IS - 2

ER -