|
|
6度边本原双本原图的分类
|
Abstract:
如果
是一个6度连通的二部图，其中
，令
，若
中存在一个指数为2的正规子群
，且
在两个分部
和
上作用是本原的，则我们称图
是
-双本原的。本文通过对6度边本原图的分类，确定了在非忠实的作用下，6度图中存在的双本原图只有
，并进一步了解6度双本原图的结构和性质。
If
is a 6-valent connected bipartite graph, where
and
, if here exists a normal subgroup
of
| [1] | 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 |
| [2] | Alspach, B. and Sutcliffe, R.J. (1979) Vertex‐Transitive Graphs of Order 2p. Annals of the New York Academy of Sciences, 319, 18-27. https://doi.org/10.1111/j.1749-6632.1979.tb32769.x |
| [3] | Chao, C. (1971) On the Classification of Symmetric Graphs with a Prime Number of Vertices. Transactions of the American Mathematical Society, 158, 247. https://doi.org/10.2307/1995785 |
| [4] | 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 |
| [5] | 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 |
| [6] | 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 |
| [7] | Hua, X., 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 |
| [8] | Hua, X. and Chen, L. (2018) Valency Seven Symmetric Graphs of Order 2pq. Czechoslovak Mathematical Journal, 68, 581-599. https://doi.org/10.21136/cmj.2018.0530-15 |
| [9] | 蓝婷, 凌波. 4p阶11度对称图[J]. 数学理论与应用, 2021, 41(2): 76. |
| [10] | Weiss, R.M. (1973) Kantenprimitive Graphen vom Grad drei. Journal of Combinatorial Theory, Series B, 15, 269-288. https://doi.org/10.1016/0095-8956(73)90041-5 |
| [11] | Giudici, M. and Li, C.H. (2010) On Finite Edge-Primitive and Edge-Quasiprimitive Graphs. Journal of Combinatorial Theory, Series B, 100, 275-298. https://doi.org/10.1016/j.jctb.2009.09.001 |
| [12] | Guo, S., Feng, Y. and Li, C.H. (2015) Edge-Primitive Tetravalent Graphs. Journal of Combinatorial Theory, Series B, 112, 124-137. https://doi.org/10.1016/j.jctb.2014.12.004 |
| [13] | Guo, S., Feng, Y. and Li, C.H. (2012) The Finite Edge-Primitive Pentavalent Graphs. Journal of Algebraic Combinatorics, 38, 491-497. https://doi.org/10.1007/s10801-012-0412-y |
| [14] | Wu, C. and Pan, J. (2020) Finite Hexavalent Edge-Primitive Graphs. Applied Mathematics and Computation, 378, Article ID: 125207. https://doi.org/10.1016/j.amc.2020.125207 |
| [15] | Ivanov, A. and Iofinova, M.E. (1985) Bi-Primitive Cubic Graphs. In: Faradžev, I.A., Ivanov, A.A., Klin, M.H. and Woldar, A.J., Eds., Investigations in Algebraic Theory of Combinatorial Objects, Springer, 459-472. |
| [16] | Li, C.H. and Zhang, H. (2014) Finite Vertex-Biprimitive Edge-Transitive Tetravalent Graphs. Discrete Mathematics, 317, 33-43. https://doi.org/10.1016/j.disc.2013.11.006 |
| [17] | Cai, Q. and Lu, Z.P. (2024) Biprimitive Edge-Transitive Pentavalent Graphs. Discrete Mathematics, 347, Article ID: 113871. https://doi.org/10.1016/j.disc.2024.113871 |
| [18] | 徐明曜. 有限群初步[M]. 北京: 科学出版社, 2014: 48-50. |
| [19] | 徐明曜. 有限群导引(上) [M]. 北京: 科学出版社, 1999: 32-37. |
| [20] | Praeger, C.E. (1990) The Inclusion Problem for Finite Primitive Permutation Groups. Proceedings of the London Mathematical Society, 3, 68-88. https://doi.org/10.1112/plms/s3-60.1.68 |
| [21] | Wu, C. and Yin, F. (2024) A Complete Classification of Edge-Primitive Graphs of Valency 6. Discrete Mathematics, 347, Article ID: 114205. https://doi.org/10.1016/j.disc.2024.114205 |
