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Normal Cayley graphs of finite groups

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Abstract

LetG be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di) graph X = Cay(G, S) of G with respect to S is defined byV (X)=G, E (X)={(g,sg)|g∈G, s∈S} A Cayley (di) graph X = Cay (G,S) is said to be normal ifR(G) ◃A = Aut (X). A group G is said to have a normal Cayley (di) graph if G has a subset S such that the Cayley (di) graph X = Cay (G, S) is normal. It is proved that every finite group G has a normal Cayley graph unlessG≅ℤ4×ℤ2 orG≅Q 8×ℤ r2 (r⩾0) and that every finite group has a normal Cayley digraph, where Zm is the cyclic group of orderm and Q8 is the quaternion group of order 8.

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Authors and Affiliations

  1. Dapartment of System Science and Mathematics, Zhengzhou University, 450052, Zhengzhou, China

    Changqun Wang

  2. Department of Applied Mathematics, Tsinghua University, 100084, Beijing, China

    Dianjun Wang

  3. Department of Mathematics, Peking University, 100871, Beijing, China

    Mingyao Xu

Authors
  1. Changqun Wang
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  2. Dianjun Wang
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  3. Mingyao Xu
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Project supported by the National Natural Science Foundation of China (Grant No. 10231060) and the Doctorial Program Foundation of Institutions of Higher Education of China.

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Wang, C., Wang, D. & Xu, M. Normal Cayley graphs of finite groups. Sci. China Ser. A-Math. 41, 242–251 (1998). https://doi.org/10.1007/BF02879042

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  • DOI: https://doi.org/10.1007/BF02879042

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