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Automorphism Groups of Circulant Digraphs With Applications to Semigroup Theory

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Abstract

We characterize the automorphism groups of circulant digraphs whose connection sets are relatively small, and of unit circulant digraphs. For each class, we either explicitly determine the automorphism group or we show that the graph is a “normal” circulant, so the automorphism group is contained in the normalizer of a cycle. Then we use these characterizations to prove results on the automorphisms of the endomorphism monoids of those digraphs. The paper ends with a list of open problems on graphs, number theory, groups and semigroups.

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Authors and Affiliations

  1. Universidade Aberta and CEMAT-Ciências Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisboa, Portugal

    João Araújo

  2. School of Mathematics and Physical Sciences, University of Hull, Kingston upon Hull, HU6 7RX, UK

    Wolfram Bentz

  3. Department of Mathematics and Statistics, Mississippi State University PO Drawer MA, Mississippi State, MS, 39762, USA

    Edward Dobson

  4. IAM, University of Primorska, Koper, 6000, Slovenia

    Edward Dobson

  5. Department of Mathematics, University of Mary Washington, Fredericksburg, VA, 22408, USA

    Janusz Konieczny

  6. Department of Math and CS, University of Lethbridge, Lethbridge, AB, T1K 3M4, Canada

    Joy Morris

Authors
  1. João Araújo
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  2. Wolfram Bentz
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  3. Edward Dobson
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  4. Janusz Konieczny
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  5. Joy Morris
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Correspondence to Wolfram Bentz.

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Araújo, J., Bentz, W., Dobson, E. et al. Automorphism Groups of Circulant Digraphs With Applications to Semigroup Theory. Combinatorica 38, 1–28 (2018). https://doi.org/10.1007/s00493-016-3403-0

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  • DOI: https://doi.org/10.1007/s00493-016-3403-0

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