On the automorphism groups of almost all circulant graphs and digraphs

Soumya Bhoumik, Ted Dobson, Joy Morris


We attempt to determine the structure of the automorphism group of a generic circulant graph. We first show that almost all circulant graphs have automorphism groups as small as possible. The second author has conjectured that almost all of the remaining circulant (di)graphs (those whose automorphism group is not as small as possible) are normal circulant (di)graphs. We show this conjecture is not true in general, but is true if we consider only those circulant (di)graphs whose order is in a "large" subset of integers. We note that all non-normal circulant (di)graphs can be classified into two natural classes (generalized wreath products, and deleted wreath type), and show that neither of these classes contains almost every non-normal circulant digraph.


Circulant graph, automorphism group, Cayley graph, DRR, GRR.

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DOI: https://doi.org/10.26493/1855-3974.315.868

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications