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Pentavalent symmetric graphs of order 2p 3

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Abstract

A graph is symmetric or 1-regular if its automorphism group is transitive or regular on the arc set of the graph, respectively. We classify the connected pentavalent symmetric graphs of order 2p 3 for each prime p. All those symmetric graphs appear as normal Cayley graphs on some groups of order 2p 3 and their automorphism groups are determined. For p = 3, no connected pentavalent symmetric graphs of order 2p 3 exist. However, for p = 2 or 5, such symmetric graph exists uniquely in each case. For p ≥ 7, the connected pentavalent symmetric graphs of order 2p 3 are all regular covers of the dipole Dip5 with covering transposition groups of order p 3, and they consist of seven infinite families; six of them are 1-regular and exist if and only if 5 | (p-1), while the other one is 1-transitive but not 1-regular and exists if and only if 5 | (p± 1). In the seven infinite families, each graph is unique for a given order.

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  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    DaWei Yang & YanQuan Feng

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  1. DaWei Yang
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  2. YanQuan Feng
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Correspondence to YanQuan Feng.

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Yang, D., Feng, Y. Pentavalent symmetric graphs of order 2p 3 . Sci. China Math. 59, 1851–1868 (2016). https://doi.org/10.1007/s11425-016-5146-1

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