一类3度弧正则Cayley图
A Family of Arc-Regular Cubic Cayley Graphs
DOI: 10.12677/PM.2022.126110, PDF,    科研立项经费支持
作者: 赖子峰：云南财经大学，统计与数学学院，云南 昆明
关键词: 弧正则图；拟本原置换群；自同构群；Arc-Regular Graph； Quasiprimitive Permutation Group； Automorphism Group
摘要: 称一个图是弧正则的，如果其全自同构群在其弧集上的作用是正则的。Xu引出问题：是否存在一个3度弧正则图，其全自同构群是不可解的。本文我们运用群论的相关知识，构造了一类3度弧正则Cayley图，并决定了此类图的全自同构群，从而得到了满足上述问题的3度弧正则图。
Abstract: A graph is called arc-regular, if its automorphism group acts regularly on its arc set. Xu introduces a question: whether there is an arc-regular cubic graph with an insolvable automorphism group. In this paper, we apply the related knowledge of group theory to construct a family of arc-regular cubic Cayley graphs and determine its automorphism group, consequently obtaining arc-regular cubic graphs that satisfy the above problems.
文章引用：赖子峰. 一类3度弧正则Cayley图[J]. 理论数学, 2022, 12(6): 1006-1010. https://doi.org/10.12677/PM.2022.126110

参考文献

[1] 徐明曜. 有限群导引(上) [M]. 北京: 科学出版社, 1999.
[2] 徐明曜. 有限群导引(下) [M]. 北京: 科学出版社, 2001.
[3] Pan, J.M. (2009) Locally Primitive Normal Cayley Graphs of Metacyclic Groups. The Electronic Journal of Combinatorics, 16, Article No. R96. [Google Scholar] [CrossRef] 
[4] Frucht, R. (1952) A One-Regular Graph of Degree Three. Canadian Journal of Mathematics, 4, 240-247. [Google Scholar] [CrossRef] 
[5] Marušič, D. and Xu, M.Y. (1997) A 1/2-Transitive Graph of Va-lency 4 with a Nonsolvable Group of Automorphisms. Journal of Graph Theory, 25, 133-138. [Google Scholar] [CrossRef] 
[6] Marušič, D. (1997) A Family of One-Regular Graphs of Valency 4. European Journal of Combinatorics, 18, 59-64. [Google Scholar] [CrossRef] 
[7] Marušič, D. and Pisanski, T. (2000) Symmetries of Hexagonal Mo-lecular Graphs on the Torus. Croatica Chemica Acta, 73, 969-981.
[8] Fang, X.G., Wang, J. and Xu, M.Y. (2002) On 1-Arc-Regular Graphs. European Journal of Combinatorics, 23, 785-791. [Google Scholar] [CrossRef] 
[9] Kwak, J.H. and Oh, J.M. (2006) One-Regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer. Acta Mathematica Sinica (English Series), 22, 1305-1320. [Google Scholar] [CrossRef] 
[10] Conder, M. and Feng, Y.Q. (2012) Arc-Regular Cubic Graphs of Order Four Times an Odd Integer. Journal of Algebraic Combinatorics, 36, 21-31. [Google Scholar] [CrossRef] 
[11] Feng, Y.Q. and Kwak, J.H. (2004) One-Regular Cubic Graphs of Order a Small Number Times a Prime or Prime Square. Journal of the Australian Mathematical Society, 76, 345-356. [Google Scholar] [CrossRef] 
[12] Zhou, J.X. and Feng, Y.Q. (2009) Tetravalent One-Regular Graphs of Order 2pq. Journal of Algebraic Combinatorics, 29, Article No. 457. [Google Scholar] [CrossRef] 
[13] 丁梦琳. 平方自由阶素数度2-弧正则图[J]. 应用数学进展, 2018, 7(4): 369-373. [Google Scholar] [CrossRef] 
[14] Wang, G. and Gao, B. (2021) Finite Two-Arc-Regular Graphs Admitting an almost Simple Group. Journal of Mathematical Research with Applications, 41, 7-13. [Google Scholar] [CrossRef] 
[15] Feng, Y.Q. and Li, Y.T. (2011) One-Regular Graphs of Square-Free Order of Prime Valency. European Journal of Combinatorics, 32, 261-275. [Google Scholar] [CrossRef] 
[16] Kwak, J.H., Kwon, Y.S. and Oh, J. (2008) Infinitely Many One-Regular Cayley Graphs on Dihedral Groups of Any Prescribed Valency. Journal of Combinatorial Theory, Series B, 98, 585-598. [Google Scholar] [CrossRef] 
[17] Kwak, J.H. and Oh, J.M. (2004) Infinitely Many Finite One-Regular Graphs of Any Even Valency. Journal of Combinatorial Theory, Series B, 90, 185-191. [Google Scholar] [CrossRef] 
[18] Pan, J.M., Huang, Z.H. and Peng, S.Q. (2016) On Arc-Regular Frobenius Metacirculants. Bulletin of the Australian Mathematical Society, 94, 20-29. [Google Scholar] [CrossRef] 
[19] Wang, C.Q. and Xu, M.Y. (2006) Non-Normal One-Regular and 4-Valent Cayley Graphs of Dihedral Groups D2n. European Journal of Combinatorics, 27, 750-766. [Google Scholar] [CrossRef] 
[20] Nuzhin, Y.N. (1992) Generating Triples of Involutions of Alter-nating Groups. Mathematical Notes, 51, 389-392. [Google Scholar] [CrossRef] 
[21] Lucchini, A. and Tamburini, M.C. (1999) Classical Groups of Large Rank as Hurwitz Groups. Journal of Algebra, 219, 531-546. [Google Scholar] [CrossRef] 
[22] Xu, M.Y. (1998) Automorphism Groups and Isomorphisms of Cayley Graphs. Discrete Mathematics, 182, 309-319. [Google Scholar] [CrossRef] 
[23] Tutte, W.T. (1947) A Family of Cubical Graphs. Mathe-matical Proceedings of the Cambridge Philosophical Society, 43, 459-474. [Google Scholar] [CrossRef] 
[24] Scott, L.L. (1980) Representations in Characteristic p. Pro-ceedings of Symposia in Pure Mathematics, 37, 319-331. [Google Scholar] [CrossRef] 
[25] Liebeck, M.W., Praeger, C.E. and Saxl, J. (1987) A Classification of the Maximal Subgroups of the Finite Alternating and Symmetric Groups. Journal of Algebra, 111, 365-383. [Google Scholar] [CrossRef] 
[26] 徐明曜. 有限群导引(上) [M]. 北京: 科学出版社, 1999.
[27] 徐明曜. 有限群导引(下) [M]. 北京: 科学出版社, 2001.
[28] Pan, J.M. (2009) Locally Primitive Normal Cayley Graphs of Metacyclic Groups. The Electronic Journal of Combinatorics, 16, Article No. R96. [Google Scholar] [CrossRef] 
[29] Frucht, R. (1952) A One-Regular Graph of Degree Three. Canadian Journal of Mathematics, 4, 240-247. [Google Scholar] [CrossRef] 
[30] Marušič, D. and Xu, M.Y. (1997) A 1/2-Transitive Graph of Va-lency 4 with a Nonsolvable Group of Automorphisms. Journal of Graph Theory, 25, 133-138. [Google Scholar] [CrossRef] 
[31] Marušič, D. (1997) A Family of One-Regular Graphs of Valency 4. European Journal of Combinatorics, 18, 59-64. [Google Scholar] [CrossRef] 
[32] Marušič, D. and Pisanski, T. (2000) Symmetries of Hexagonal Mo-lecular Graphs on the Torus. Croatica Chemica Acta, 73, 969-981.
[33] Fang, X.G., Wang, J. and Xu, M.Y. (2002) On 1-Arc-Regular Graphs. European Journal of Combinatorics, 23, 785-791. [Google Scholar] [CrossRef] 
[34] Kwak, J.H. and Oh, J.M. (2006) One-Regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer. Acta Mathematica Sinica (English Series), 22, 1305-1320. [Google Scholar] [CrossRef] 
[35] Conder, M. and Feng, Y.Q. (2012) Arc-Regular Cubic Graphs of Order Four Times an Odd Integer. Journal of Algebraic Combinatorics, 36, 21-31. [Google Scholar] [CrossRef] 
[36] Feng, Y.Q. and Kwak, J.H. (2004) One-Regular Cubic Graphs of Order a Small Number Times a Prime or Prime Square. Journal of the Australian Mathematical Society, 76, 345-356. [Google Scholar] [CrossRef] 
[37] Zhou, J.X. and Feng, Y.Q. (2009) Tetravalent One-Regular Graphs of Order 2pq. Journal of Algebraic Combinatorics, 29, Article No. 457. [Google Scholar] [CrossRef] 
[38] 丁梦琳. 平方自由阶素数度2-弧正则图[J]. 应用数学进展, 2018, 7(4): 369-373. [Google Scholar] [CrossRef] 
[39] Wang, G. and Gao, B. (2021) Finite Two-Arc-Regular Graphs Admitting an almost Simple Group. Journal of Mathematical Research with Applications, 41, 7-13. [Google Scholar] [CrossRef] 
[40] Feng, Y.Q. and Li, Y.T. (2011) One-Regular Graphs of Square-Free Order of Prime Valency. European Journal of Combinatorics, 32, 261-275. [Google Scholar] [CrossRef] 
[41] Kwak, J.H., Kwon, Y.S. and Oh, J. (2008) Infinitely Many One-Regular Cayley Graphs on Dihedral Groups of Any Prescribed Valency. Journal of Combinatorial Theory, Series B, 98, 585-598. [Google Scholar] [CrossRef] 
[42] Kwak, J.H. and Oh, J.M. (2004) Infinitely Many Finite One-Regular Graphs of Any Even Valency. Journal of Combinatorial Theory, Series B, 90, 185-191. [Google Scholar] [CrossRef] 
[43] Pan, J.M., Huang, Z.H. and Peng, S.Q. (2016) On Arc-Regular Frobenius Metacirculants. Bulletin of the Australian Mathematical Society, 94, 20-29. [Google Scholar] [CrossRef] 
[44] Wang, C.Q. and Xu, M.Y. (2006) Non-Normal One-Regular and 4-Valent Cayley Graphs of Dihedral Groups D2n. European Journal of Combinatorics, 27, 750-766. [Google Scholar] [CrossRef] 
[45] Nuzhin, Y.N. (1992) Generating Triples of Involutions of Alter-nating Groups. Mathematical Notes, 51, 389-392. [Google Scholar] [CrossRef] 
[46] Lucchini, A. and Tamburini, M.C. (1999) Classical Groups of Large Rank as Hurwitz Groups. Journal of Algebra, 219, 531-546. [Google Scholar] [CrossRef] 
[47] Xu, M.Y. (1998) Automorphism Groups and Isomorphisms of Cayley Graphs. Discrete Mathematics, 182, 309-319. [Google Scholar] [CrossRef] 
[48] Tutte, W.T. (1947) A Family of Cubical Graphs. Mathe-matical Proceedings of the Cambridge Philosophical Society, 43, 459-474. [Google Scholar] [CrossRef] 
[49] Scott, L.L. (1980) Representations in Characteristic p. Pro-ceedings of Symposia in Pure Mathematics, 37, 319-331. [Google Scholar] [CrossRef] 
[50] Liebeck, M.W., Praeger, C.E. and Saxl, J. (1987) A Classification of the Maximal Subgroups of the Finite Alternating and Symmetric Groups. Journal of Algebra, 111, 365-383. [Google Scholar] [CrossRef] 

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