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Pure Mathematics 2025
一类几乎单型的拟本原图的刻画
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Abstract:
设
是一个连通图，
，
是
-边传递但不是
-弧传递的。在奇数阶2倍素数度图的研究基础上，本文聚焦于拟本原非2-弧传递的情况，通过研究几乎单群
作用在
上的拟本原情形，对奇素数幂阶2倍素数度非2-弧传递图展开刻画。研究发现，此类图的结构较为特殊，要么是完全图
或
，要么同构于一个27阶10度图。这一结论进一步丰富了图论中关于特殊度数和传递性图的分类成果，为后续相关研究提供了重要参考。
Let
be a connected graph,
, and
be
-edge-transitive but not
-arc-transitive. Based on the research of graphs with odd order and twice prime valency, this paper focuses on the quasiprimitive
| [1] | 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 |
| [2] | Praeger, C.E. (1997) Finite Quasiprimitive Graphs. In: Bailey, R.A., Ed., Surveys in Combinatorics, Cambridge University Press, 65-86. https://doi.org/10.1017/cbo9780511662119.005 |
| [3] | Frucht, R. (1939) Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compositio Mathematica, 6, 239-250. |
| [4] | 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 |
| [5] | Weiss, R. (1981) The Nonexistence of 8-Transitive Graphs. Combinatorica, 1, 309-311. https://doi.org/10.1007/bf02579337 |
| [6] | 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 |
| [7] | Gui Fang, X. and Praeger, C.E. (1999) Finite Two-Are Transitive Graphs Admitting a REE Simple Group. Communications in Algebra, 27, 3755-3769. https://doi.org/10.1080/00927879908826660 |
| [8] | Gui Fang, X. and Preager, C.E. (1999) Fintte Two-Arc Transitive Graphs Admitting a Suzuki Simple Group. Communications in Algebra, 27, 3727-3754. https://doi.org/10.1080/00927879908826659 |
| [9] | Li, C.H. and Pan, J. (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 |
| [10] | Li, C.H., Li, J.J. and Lu, Z.P. (2023) Two-Arc-Transitive Graphs of Odd Order: I. Journal of Algebraic Combinatorics, 57, 1253-1264. https://doi.org/10.1007/s10801-023-01224-8 |
| [11] | Li, C.H., Li, J.J. and Lu, Z.P. (2021) Two-Arc-Transitive Graphs of Odd Order—II. European Journal of Combinatorics, 96, Article ID: 103354. https://doi.org/10.1016/j.ejc.2021.103354 |
| [12] | Guo, S., Li, Y. and Hua, X. (2016) (G, S)-Transitive Graphs of Valency 7. Algebra Colloquium, 23, 493-500. https://doi.org/10.1142/s100538671600047x |
| [13] | Zhou, J. and Feng, Y. (2010) On Symmetric Graphs of Valency Five. Discrete Mathematics, 310, 1725-1732. https://doi.org/10.1016/j.disc.2009.11.019 |
| [14] | Zhou, J. (2009) Tetravalent s-Transitive Graphs of Order 4p. Discrete Mathematics, 309, 6081-6086. https://doi.org/10.1016/j.disc.2009.05.014 |
| [15] | Praeger, C.E. and Xu, M. (1989) A Characterization of a Class of Symmetric Graphs of Twice Prime Valency. European Journal of Combinatorics, 10, 91-102. https://doi.org/10.1016/s0195-6698(89)80037-x |
| [16] | Gorenstein, D. (1982) Finite Simple Groups. Plenum Press. |
| [17] | Liao, H.C., Li, J.J. and Lu, Z.P. (2020) On Quasiprimitive Edge-Transitive Graphs of Odd Order and Twice Prime Valency. Journal of Group Theory, 23, 1017-1037. https://doi.org/10.1515/jgth-2019-0091 |
| [18] | Guralnick, R.M. (1983) Subgroups of Prime Power Index in a Simple Group. Journal of Algebra, 81, 304-311. https://doi.org/10.1016/0021-8693(83)90190-4 |
| [19] | Li, C.H., Pan, J. and Ma, L. (2009) Locally Primitive Graphs of Prime-Power Order. Journal of the Australian Mathematical Society, 86, 111-122. https://doi.org/10.1017/s144678870800089x |
| [20] | 王杰. 典型群引论[M]. 北京: 北京大学出版社, 2015. |
