18pq阶5度对称图的分类
A Classification of Pentavalent Symmetric Graphs on 18pq Vertices
DOI: 10.12677/pm.2026.1610227, PDF,   
作者: 何古落*, 凌 波：云南民族大学数学与计算机科学学院，云南 昆明
关键词: 对称图；自同构群；正规商图；Symmetric Graph； Automorphism Group； Normal Quotient
摘要: 若一个图的自同构群在其弧集上是传递作用的，则称该图为对称图。本文运用群论方法，对所有可容许素数对 ( p,q ) ，完成了阶为 18pq 的连通五度对称图的分类。确定了这类图的自同构群均为特定的有限几乎单群，拓展了复合阶五度对称图已有的分类结果。
Abstract: A graph is symmetric if its automorphism group acts transitively on its arcs. Using group-theoretic methods, we classify all connected pentavalent symmetric graphs of order 18pq for each admissible pair ( p,q ) . The automorphism groups are determined as specific finite almost simple groups, extending known classifications of pentavalent symmetric graphs with composite orders.
文章引用：何古落, 凌波. 18pq阶5度对称图的分类[J]. 理论数学, 2026, 16(10): 226-236. https://doi.org/10.12677/pm.2026.1610227

参考文献

[1] Tutte, W.T. (1947) A Family of Cubical Graphs. Mathematical Proceedings of the Cambridge Philosophical Society, 43, 459-474.
https://doi.org/10.1017/s0305004100023720
[2] Tutte, W.T. (1959) On the Symmetry of Cubic Graphs. Canadian Journal of Mathematics, 11, 621-624.
https://doi.org/10.4153/cjm-1959-057-2
[3] Chao, C. (1971) On the Classification of Symmetric Graphs with a Prime Number of Vertices. Transactions of the American Mathematical Society, 158, 247-256.
https://doi.org/10.1090/s0002-9947-1971-0279000-7
[4] Praeger, C.E., Wang, R.J. and Xu, M.Y. (1993) Symmetric Graphs of Order a Product of Two Distinct Primes. Journal of Combinatorial Theory, Series B, 58, 299-318.
https://doi.org/10.1006/jctb.1993.1046
[5] Praeger, C.E. and Xu, M.Y. (1993) Vertex-Primitive Graphs of Order a Product of Two Distinct Primes. Journal of Combinatorial Theory, Series B, 59, 245-266.
https://doi.org/10.1006/jctb.1993.1068
[6] Wang, R.J. and Xu, M.Y. (1993) A Classification of Symmetric Graphs of Order 3p. Journal of Combinatorial Theory, Series B, 58, 197-216.
https://doi.org/10.1006/jctb.1993.1037
[7] Li, C.H. (2001) On Finite S-Transitive Graphs of Odd Order. Journal of Combinatorial Theory, Series B, 81, 307-317.
https://doi.org/10.1006/jctb.2000.2012
[8] Conder, M. and Dobcsányi, P. (2002) Trivalent Symmetric Graphs on up to 768 Vertices. The Journal of Combinatorial Mathematics and Combinatorial Computing, 40, 41-63.
[9] Zhou, J.X. and Feng, Y.Q. (2009) Tetravalent One-Regular Graphs of Order 2pq. Journal of Algebraic Combinatorics, 29, 457-471.
https://doi.org/10.1007/s10801-008-0146-z
[10] Zhou, J.X. and Feng, Y.Q. (2010) On Symmetric Graphs of Valency Five. Discrete Mathematics, 310, 1725-1732.
https://doi.org/10.1016/j.disc.2009.11.019
[11] Alaeiyan, M. and Akbarizadeh, M. (2019) Classification of the Pentavalent Symmetric Graphs of Order 18p. Indian Journal of Pure and Applied Mathematics, 50, 485-497.
https://doi.org/10.1007/s13226-019-0340-9
[12] Guo, S.T., Zhou, J.X. and Feng, Y.Q. (2011) Pentavalent Symmetric Graphs of Order 12p. The Electronic Journal of Combinatorics, 18, P233.
https://doi.org/10.37236/720
[13] Hua, X.H. and Feng, Y.Q. (2011) Pentavalent Symmetric Graphs of Order 8p. Journal of Beijing Jiaotong University, 35, 132-135, 141. (In Chinese)
[14] Hua, X.H., Feng, Y. and Lee, J. (2011) Pentavalent Symmetric Graphs of Order 2pq. Discrete Mathematics, 311, 2259-2267.
https://doi.org/10.1016/j.disc.2011.07.007
[15] Ling, B. (2017) Classifying Pentavalent Symmetric Graphs of Order 24p. Bulletin of the Iranian Mathematical Society, 43, 1855-1866.
[16] Ling, B., Wu, C.X. and Lou, B.G. (2014) Pentavalent Symmetric Graphs of Order 30p. Bulletin of the Australian Mathematical Society, 90, 353-362.
https://doi.org/10.1017/s0004972714000616
[17] Pan, J., Lou, B. and Liu, C. (2013) Arc-Transitive Pentavalent Graphs of Order 4pq. The Electronic Journal of Combinatorics, 20, P36.
https://doi.org/10.37236/2373
[18] Ding, S.Y., Ling, B., Lou, B. and Pan, J. (2016) Arc-Transitive Pentavalent Graphs of Square-Free Order. Graphs and Combinatorics, 32, 2355-2366.
https://doi.org/10.1007/s00373-016-1717-8
[19] Li, Y.T. and Feng, Y.Q. (2010) Pentavalent One-Regular Graphs of Square-Free Order. Algebra Colloquium, 17, 515-524.
https://doi.org/10.1142/s1005386710000490
[20] Meymandi, M.A., Alaeiyan, M. and Scapellato, R. (2022) Classification of the Pentavalent Symmetric Graphs of Order 8pq. International Journal of Group Theory, 11, 259-270.
[21] Dickson, L.E. (1958) Linear Groups with an Exposition of the Galois Field Theory. Dover.
[22] Robinson, D.J. (1982) A Course in the Theory of Groups. Springer-Verlag.
[23] The GAP Group, GAP-Groups (2024) Algorithms, and Programming, Version 4.13.1.
https://www.gap-system.org/
[24] Bosma, W., Cannon, J. and Playoust, C. (1997) The MAGMA Algebra System I: The User Language. Journal of Symbolic Computation, 24, 235-265.
https://doi.org/10.1006/jsco.1996.0125
[25] Cameron, P.J., Omidi, G.R. and Tayfeh-Rezaie, B. (2006) 3-Designs from PGL(2,q). The Electronic Journal of Combinatorics, 13, R50.
https://doi.org/10.37236/1076
[26] Guo, S. and Feng, Y. (2012) A Note on Pentavalent s-Transitive Graphs. Discrete Mathematics, 312, 2214-2216.
https://doi.org/10.1016/j.disc.2012.04.015
[27] Li, C.H. and Pan, J.M. (2008) Finite 2-Arc-Transitive Abelian Cayley Graphs. European Journal of Combinatorics, 29, 148-158.
https://doi.org/10.1016/j.ejc.2006.12.001
[28] Praeger, C.E. (1993) An O’nan-Scott Theorem for Finite Quasiprimitive Permutation Groups and an Application to 2-Arc Transitive Graphs. Journal of the London Mathematical Society, 2, 227-239.
https://doi.org/10.1112/jlms/s2-47.2.227
[29] Sabidussi, G. (1964) Vertex-Transitive Graphs. Monatshefte für Mathematik, 68, 426-438.
https://doi.org/10.1007/bf01304186
[30] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A. and Wilson, R.A. (1985) Atlas of Finite Groups. Oxford University Press.
[31] Gorenstein, D. (1982) Finite Simple Groups. Plenum Press.

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