Automorphism groups of quartic bicirculant nut graphs

Characterize the possible full automorphism groups of quartic bicirculant nut graphs, extending the results known for special cases.

Background

The paper notes that several quartic bicirculant nut graphs of the same order can have mutually nonisomorphic full automorphism groups. Although some results on automorphism groups of quartic bicirculant graphs are available in special cases, the general classification or characterization remains unresolved. This problem concerns determining which automorphism groups can occur for quartic bicirculant nut graphs and, more broadly, for quartic bicirculant graphs.

References

It would be interesting to study the possible automorphism groups of QBN graphs. For example, the five QBN graphs B2(10; 1, 2, 1), B2(10; 1, 1, 1), B2(10; 1, 2,5), B2(10; 1, 3,5) and B1(10; 2, 2) are of the same order and have mutually nonisomorphic (full) automorphism groups. The general question about the automorphism groups of quartic bicirculant graphs remains largely open, while some results have been obtained in special cases; see [15, 16, 17, 18, 26].

— Classification of quartic bicirculant nut graphs  (2502.06353 - Damnjanović et al., 10 Feb 2025) in Section 6 (Conclusion)