## Abstract

Given a graph G with n vertices and an Abelian group A of order n, an A-distance antimagic labelling of G is a bijection from V (G) to A such that the vertices of G have pairwise distinct weights, where the weight of a vertex is the sum (under the operation of A) of the labels assigned to its neighbours. An A-distance magic labelling of G is a bijection from V (G) to A such that the weights of all vertices of G are equal to the same element of A. In this paper we study these new labellings under a general setting with a focus on product graphs. We prove among other things several general results on group antimagic or magic labellings for Cartesian, direct and strong products of graphs. As applications we obtain several families of graphs admitting group distance antimagic or magic labellings with respect to elementary Abelian groups, cyclic groups or direct products of such groups.

Original language | English (US) |
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Pages (from-to) | 1159-1176 |

Number of pages | 18 |

Journal | Acta Mathematica Sinica, English Series |

Volume | 32 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1 2016 |

Externally published | Yes |

### Bibliographical note

Funding Information:1 Partially supported by the Polish Ministry of Science and Higher Education. 2 Email: cichacz@agh.edu.pl 3 Email: dalibor@d.umn.edu 4 Email: kiki@sci.ui.ac.id 5 Supported by the Australian Research Council. 6 Email: smzhou@ms.unimelb.edu.au

Publisher Copyright:

© 2016, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.

## Keywords

- Distance magic labelling
- distance antimagic labelling
- group labelling