Abstract
Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph (Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph (also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits (of equal size). In this paper, we introduce the concept of SCI-graph (semi-Cayley isomorphism) and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.
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References
Arezoomand, M., Taeri, B.: On the characteristic polynomial of n-Cayley digraphs. Electron. J. Combin., 20(3), Paper57, 14pp (2013)
Babai, L.: Isomorphism problem for a class of point-symmetric structures. Acta Math. Acad. Sci. Hungar., 29, 329–336 (1977)
Berkovich, Y.: Groups of Prime Power Order, Volume 1, Walter de Gruyter, Berlin, 2008
Dixon, J. D., Mortimer, B.: Permutation groups, Graduate Texts in Mathematics 163, Springer-Verlag, New York, 1996
Harary, F.: Graph Theory, Addison-Welsey, Reading, Massachusetts, 1969
Jin, W., Liu, W. J.: Two results on BCI-subset of finite groups. Ars Combin., 93, 169–173 (2009)
Jin, W., Liu, W. J.: A classification of nonabelian simple 3-BCI-groups. European J. Combin., 31, 1257–1264 (2010)
Jin, W., Liu, W. J.: On Sylow subgroups of BCI-groups. Util. Math., 86, 313–320 (2011)
Koike, H., Kovács, I.: Isomorphic tetravalent cyclic Haar graphs. Ars Math. Contemp., 7(1), 215–235 (2014)
Kovács, I., Malnič, I., Marušič, D., et al.: One-Matching bi-Cayley graphs over abelian groups. European J. Combin., 30, 602–616 (2009)
Li, C. H.: On isomorphisms of finite Cayley graphs-a survay. Discrete Math., 256, 301–334 (2002)
Lu, Z. P., Wang, C. Q., Xu, M. Y.: Semisymmetric cubic graphs constructed from bi-Cayley graphs of An. Ars Combin., 80, 177–187 (2006)
Malnič, A., Marušič, D., Šparl, P., et al.: Symmetry structure of bicirculants. Discrete Math., 307, 409–414 (2007)
de Resmini, M. J., Jungnickel, D.: Strongly regular semi-Cayley graphs. J. Algebraic Combin., 1, 217–228 (1992)
Xu, S. J., Jin, W., Shi, Q., et al.: The BCI-property of the bi-Cayley graphs. J. Guangxi Norm. Univ.: Nat. Sci. Edition, 26, 33–36 (2008)
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Arezoomand, M., Taeri, B. Isomorphisms of finite semi-Cayley graphs. Acta. Math. Sin.-English Ser. 31, 715–730 (2015). https://doi.org/10.1007/s10114-015-4356-8
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DOI: https://doi.org/10.1007/s10114-015-4356-8