Abstract
In this paper, we define meta-Cayley graphs on dihedral groups. We fully determine the automorphism groups of the constructed graphs in question. Further, we prove that some of the graphs that we have constructed do not admit subgroups which act regularly on their vertex set; thus proving that they cannot be represented as Cayley graphs on groups.

Similar content being viewed by others
References
Alspach, B., Parsons, T.D.: A construction for vertex transitive graphs. Can. J. Math. 34, 307–318 (1982)
Cheng, H., Ghasemi, M., Qiao, S.: Tetravalent vertex-transitive graphs of order twice a prime square. Graphs Comb. 32, 1763–1771 (2016)
Frucht, R., Graver, J.E., Watkins, M.E.: The groups of the generalized Petersen graphs. Math. Proc. Camb. Philos. Soc. 70, 211–218 (1971)
Gauyacq, G.: On quasi-Cayley graphs. Discrete Appl. Math. 77, 43–58 (1997)
Godsil, C.D.: More odd graph theory. Discrete Math. 32, 205–217 (1980)
Ivanov, A.A., Praeger, C.E.: Problem session at ALCOM-91. Eur. J. Comb. 15, 105–112 (1994)
Kantor, W.M.: Primitive permutation groups of odd degree with an application to projective planes. J. Algebra 106, 15–45 (1987)
Marušič, D.: Cayley properties of vertex symmetric graphs. Ars Comb. 16B, 297–302 (1983)
Marušič, D., Scapellato, R.: Characterising vertex-transitive $pq$-graphs with an imprimitive automorphism group. J. Graph Theory 16, 375–387 (1992)
Marušič, D., Scapellato, R.: Imprimitive representations of $SL(2,2^k)$. J. Comb. Theory (B) 58, 46–57 (1993)
Miller, G.A.: Automorphisms of the dihedral groups. Proc. Natl. Acad. Sci. USA 28(9), 368–371 (1942)
Mwambene, E.: Representing Graphs on Groupoids: Symmetry and Form. PhD Thesis, University of Vienna (2001)
Mwambene, E.: Representing vertex-transitive graphs on groupoids. Quaest. Math. 29(3), 279–284 (2006)
Mwambene, E.: Cayley graphs on left quasi-groups and groupoids representing k-dimensional generalised Petersen graphs. Discrete Math. 309, 2544–2547 (2009)
Mwambene, E.: On non-Cayley vertex-transitive graphs and the meta-Cayley graphs. Quaest. Math. 34(4), 425–431 (2011)
Praeger, C.E., Xu, M.Y.: Vertex-transitive graphs of order a product of two distinct primes. J. Comb. Theory (B) 59, 245–266 (1993)
Sabidussi, G.: On a class of fixed-point-free graphs. Proc. Am. Math. Soc. 9(5), 800–804 (1958)
Watkins, M.E.: Vertex-transitive graphs that are not Cayley graphs. In: Hahn, G., et al. (eds.) Cycles and Rays, pp. 243–256. Kluwer, Alphen aan den Rijn (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Allie, I., Mwambene, E. Some Meta-Cayley Graphs on Dihedral Groups. Graphs and Combinatorics 35, 1433–1446 (2019). https://doi.org/10.1007/s00373-019-02097-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-019-02097-0