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Hua, XiaoHui; Feng, YanQuan

Cubic semisymmetric graphs of order \(8p ^{3}\). (English) Zbl 1237.05052

Sci. China, Math. 54, No. 9, 1937-1949 (2011).
Summary: A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. By J. Folkman [J. Comb. Theory 3, 215–232 (1967; Zbl 0158.42501)], there is no semisymmetric graph of order \(2p\) or \(2p ^{2}\) for a prime \(p\), and by A. Malnič et al. [Discrete Math. 307, No. 17–18, 2156–2175 (2007; Zbl 1136.05026)], there exists a unique cubic semisymmetric graph of order \(2p ^{3}\), the so called Gray graph of order 54. In this paper, it is shown that there is no connected cubic semisymmetric graph of order \(4p ^{3}\) and that there exists a unique cubic semisymmetric graph of order \(8p ^{3}\), which is a \(\mathbb Z_{2} \times \mathbb Z_{2}\)-covering of the Gray graph.

Cited in 3 Documents

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures

Keywords:

edge-transitive graph; semisymmetric graph; regular covering; Gray graph

Citations:

Zbl 0158.42501; Zbl 1136.05026

Software:

Magma
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Full Text: DOI

References:

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