| Title:
|
Valency seven symmetric graphs of order $2pq$ (English) |
| Author:
|
Hua, Xiao-Hui |
| Author:
|
Chen, Li |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
581-599 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, all connected valency seven symmetric graphs of order $2pq$ are classified, where $p$, $q$ are distinct primes. It follows from the classification that there is a unique connected valency seven symmetric graph of order $4p$, and that for odd primes $p$ and $q$, there is an infinite family of connected valency seven one-regular graphs of order $2pq$ with solvable automorphism groups, and there are four sporadic ones with nonsolvable automorphism groups, which is $1,2,3$-arc transitive, respectively. In particular, one of the four sporadic ones is primitive, and the other two of the four sporadic ones are bi-primitive. (English) |
| Keyword:
|
arc-transitive graph |
| Keyword:
|
symmetric graph |
| Keyword:
|
$s$-regular graph |
| MSC:
|
05C25 |
| MSC:
|
20B25 |
| idZBL:
|
Zbl 06986958 |
| idMR:
|
MR3851877 |
| DOI:
|
10.21136/CMJ.2018.0530-15 |
| . |
| Date available:
|
2018-08-09T13:08:32Z |
| Last updated:
|
2020-10-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147353 |
| . |
| Reference:
|
[1] Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language.J. Symb. Comput. 24 (1997), 235-265. Zbl 0898.68039, MR 1484478, 10.1006/jsco.1996.0125 |
| Reference:
|
[2] Cheng, Y., Oxley, J.: On weakly symmetric graphs of order twice a prime.J. Comb. Theory, Ser. B 42 (1987), 196-211. Zbl 0583.05032, MR 0884254, 10.1016/0095-8956(87)90040-2 |
| Reference:
|
[3] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups.Oxford University Press, Eynsham (1985). Zbl 0568.20001, MR 0827219 |
| Reference:
|
[4] Djoković, D. Ž., Miller, G. L.: Regular groups of automorphisms of cubic graphs.J. Comb. Theory, Ser. B 29 (1980), 195-230. Zbl 0385.05040, MR 0586434, 10.1016/0095-8956(80)90081-7 |
| Reference:
|
[5] Fang, X. G., Praeger, C. E.: Finite two-arc transitive graphs admitting a Suzuki simple group.Commun. Algebra 27 (1999), 3727-3754. Zbl 0956.05049, MR 1699629, 10.1080/00927879908826659 |
| Reference:
|
[6] Fang, X., Wang, J., Xu, M. Y.: On 1-arc-regular graphs.Eur. J. Comb. 23 (2002), 785-791. Zbl 1014.05033, MR 1928997, 10.1006/eujc.2002.0579 |
| Reference:
|
[7] Feng, Y.-Q., Ghasemi, M., Yang, D.-W.: Cubic symmetric graphs of order $8p^3$.Discrete Math. 318 (2014), 62-70. Zbl 1281.05075, MR 3141628, 10.1016/j.disc.2013.11.013 |
| Reference:
|
[8] Feng, Y.-Q., Kwak, J. H.: Cubic symmetric graphs of order a small number times a prime or a prime square.J. Comb. Theory, Ser. B 97 (2007), 627-646. Zbl 1118.05043, MR 2325802, 10.1016/j.jctb.2006.11.001 |
| Reference:
|
[9] Feng, Y.-Q., Kwak, J. H., Xu, M.-Y.: Cubic $s$-regular graphs of order $2p^3$.J. Graph Theory 52 (2006), 341-352. Zbl 1100.05073, MR 2242833, 10.1002/jgt.20169 |
| Reference:
|
[10] Feng, Y.-Q., Li, Y.-T.: One-regular graphs of square-free order of prime valency.Eur. J. Comb. 32 (2011), 265-275. Zbl 1229.05114, MR 2738546, 10.1016/j.ejc.2010.10.002 |
| Reference:
|
[11] Gardiner, A., Praeger, C. E.: On 4-valent symmetric graphs.Eur. J. Comb. 15 (1994), 375-381. Zbl 0806.05037, MR 1279075, 10.1006/eujc.1994.1041 |
| Reference:
|
[12] Gardiner, A., Praeger, C. E.: A characterization of certain families of 4-valent symmetric graphs.Eur. J. Comb. 15 (1994), 383-397. Zbl 0806.05038, MR 1279076, 10.1006/eujc.1994.1042 |
| Reference:
|
[13] Gorenstein, D.: Finite Simple Groups. An Introduction to Their Classification.The University Series in Mathematics, Plenum Press, New York (1982). Zbl 0483.20008, MR 0698782, 10.1007/978-1-4684-8497-7 |
| Reference:
|
[14] Guo, S.-T., Feng, Y.-Q.: A note on pentavalent $s$-transitive graphs.Discrete Math. 312 (2012), 2214-2216. Zbl 1246.05105, MR 2926093, 10.1016/j.disc.2012.04.015 |
| Reference:
|
[15] Guo, S., Li, Y., Hua, X.: $(G,s)$-transitive graphs of valency 7.Algebra Colloq. 23 (2016), 493-500. Zbl 1345.05044, MR 3514538, 10.1142/S100538671600047X |
| Reference:
|
[16] Guo, S.-T., Shi, J., Zhang, Z.-J.: Heptavalent symmetric graphs of order $4p$.South Asian J. Math. 1 (2011), 131-136. Zbl 1242.05119, MR 3974117 |
| Reference:
|
[17] Hua, X.-H., Feng, Y.-Q., Lee, J.: Pentavalent symmetric graphs of order $2pq$.Discrete Math. 311 (2011), 2259-2267. Zbl 1246.05072, MR 2825671, 10.1016/j.disc.2011.07.007 |
| Reference:
|
[18] Li, Y., Feng, Y.-Q.: Pentavalent one-regular graphs of square-free order.Algebra Colloq. 17 (2010), 515-524. Zbl 1221.05201, MR 2660442, 10.1142/S1005386710000490 |
| Reference:
|
[19] Liebeck, M. W., Praeger, C. E., Saxl, J.: A classification of the maximal subgroups of the finite alternating and symmetric groups.J. Algebra 111 (1987), 365-383. Zbl 0632.20011, MR 0916173, 10.1016/0021-8693(87)90223-7 |
| Reference:
|
[20] Lorimer, P.: Vertex-transitive graphs: symmetric graphs of prime valency.J. Graph Theory 8 (1984), 55-68. Zbl 0535.05031, MR 0732018, 10.1002/jgt.3190080107 |
| Reference:
|
[21] McKay, B. D.: Transitive graphs with fewer than twenty vertices.Math. Comput. 33 (1979), 1101-1121. Zbl 0411.05046, MR 0528064, 10.2307/2006085 |
| Reference:
|
[22] Miller, R. C.: The trivalent symmetric graphs of girth at most six.J. Comb. Theory, Ser. B 10 (1971), 163-182. Zbl 0223.05113, MR 0285435, 10.1016/0095-8956(71)90075-X |
| Reference:
|
[23] Oh, J.-M.: A classification of cubic $s$-regular graphs of order $14p$.Discrete Math. 309 (2009), 2721-2726. Zbl 1208.05055, MR 2523779, 10.1016/j.disc.2008.06.025 |
| Reference:
|
[24] Oh, J.-M.: A classification of cubic $s$-regular graphs of order $16p$.Discrete Math. 309 (2009), 3150-3155. Zbl 1177.05052, MR 2526732, 10.1016/j.disc.2008.09.001 |
| Reference:
|
[25] Pan, J., Ling, B., Ding, S.: One prime-valent symmetric graphs of square-free order.Ars Math. Contemp. 15 (2018), 53-65. MR 3862077, 10.26493/1855-3974.1161.3b9 |
| Reference:
|
[26] Pan, J., Lou, B., Liu, C.: Arc-transitive pentavalent graphs of order $4pq$.Electron. J. Comb. 20 (2013), Researh Paper P36, 9 pages. Zbl 1266.05061, MR 3035046 |
| Reference:
|
[27] Potočnik, P.: A list of 4-valent 2-arc-transitive graphs and finite faithful amalgams of index $(4,2)$.Eur. J. Comb. 30 (2009), 1323-1336. Zbl 1208.05056, MR 2514656, 10.1016/j.ejc.2008.10.001 |
| Reference:
|
[28] Sabidussi, G.: Vertex-transitive graphs.Monatsh. Math. 68 (1964), 426-438. Zbl 0136.44608, MR 0175815, 10.1007/BF01304186 |
| Reference:
|
[29] Sims, C. C.: Graphs and finite permutation groups.Math. Z. 95 (1967), 76-86. Zbl 0244.20001, MR 0204509, 10.1007/BF01117534 |
| Reference:
|
[30] Suzuki, M.: Group Theory I.Grundlehren der Mathematischen Wissenschaften 247, Springer, Berlin (1982). Zbl 0472.20001, MR 0648772, 10.1007/978-3-642-61804-8 |
| Reference:
|
[31] Wilson, R. A.: The Finite Simple Groups.Graduate Texts in Mathematics 251, Springer, London (2009). Zbl 1203.20012, MR 2562037, 10.1007/978-1-84800-988-2 |
| Reference:
|
[32] Xu, J., Xu, M.: Arc-transitive Cayley graphs of valency at most four on abelian groups.Southeast Asian Bull. Math. 25 (2001), 355-363. Zbl 0993.05086, MR 1935107, 10.1007/s10012-001-0355-z |
| Reference:
|
[33] Xu, M.-Y.: Automorphism groups and isomorphisms of Cayley digraphs.Discrete Math. 182 (1998), 309-319. Zbl 0887.05025, MR 1603719, 10.1016/S0012-365X(97)00152-0 |
| Reference:
|
[34] Zhou, J.-X., Feng, Y.-Q.: Tetravalent $s$-transitive graphs of order twice a prime power.J. Aust. Math. Soc. 88 (2010), 277-288. Zbl 1214.05052, MR 2629936, 10.1017/S1446788710000066 |
| . |
