### Abstract

Let X be a finite connected graph, possibly with multiple edges. We provide each edge of the graph by two possible orientations. An automorphism group of a graph acts harmonically if it acts freely on the set of directed edges of the graph. Following M. Baker and S. Norine define a genus g of the graph X to be the rank of the first homology group. A finite group acting harmonically on a graph of genus g is a natural discrete analogue of a finite group of automorphisms acting on a Riemann surface of genus g. In the present paper, we give a sharp upper bound for the size of cyclic group acting harmonically on a graph of genus g≥ 2 with a given number of fixed points. Similar results, for closed orientable surfaces, were obtained earlier by T. Szemberg, I. Farkas and H. M. Kra.

Original language | English |
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Pages (from-to) | 2021-2031 |

Number of pages | 11 |

Journal | Analysis and Mathematical Physics |

Volume | 9 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Dec 2019 |

### Keywords

- Automorphism of graph
- Cyclic group
- Fixed point
- Genus of graph
- Graph covering
- Graph of groups
- Orbifold
- Riemann surface