Abstract
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are Cayley graphs of abelian groups. Such groups can be constructed using a generalization to \(\mathbb {Z}^n\) of the concept of congruence in \(\mathbb {Z}\). Here we use this approach to present some families of mixed graphs, which, for every fixed value of the degree, have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.
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Acknowledgements
The first two authors have been partially supported by the project 2017SGR1087 of the Agency for the Management of University and Research Grants (AGAUR) of the Catalan Government, and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. The first and the third authors have been supported in part by Grant MTM2017-86767-R of the Spanish Government.
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The research of C. Dalfó has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreement no. 734922.
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Dalfó, C., Fiol, M.A. & López, N. New Moore-Like Bounds and Some Optimal Families of Abelian Cayley Mixed Graphs. Ann. Comb. 24, 405–424 (2020). https://doi.org/10.1007/s00026-020-00496-2
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DOI: https://doi.org/10.1007/s00026-020-00496-2
Keywords
- Mixed graph
- Degree/diameter problem
- Moore bound
- Cayley graph
- Abelian group
- Congruences in \(\mathbb {Z}^n\)