Tetravalent Arc-Transitive Graphs of Order 3p 2
Mohsen Ghasemi
Discussiones Mathematicae Graph Theory, Tome 34 (2014), p. 567-575 / Harvested from The Polish Digital Mathematics Library

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. Let p be a prime. In this article a complete classification of tetravalent s-transitive graphs of order 3p2 is given

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:268227
@article{bwmeta1.element.doi-10_7151_dmgt_1751,
     author = {Mohsen Ghasemi},
     title = {
      Tetravalent Arc-Transitive Graphs of Order 3p
      2
    },
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {34},
     year = {2014},
     pages = {567-575},
     zbl = {1305.05101},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1751}
}
Mohsen Ghasemi. 
      Tetravalent Arc-Transitive Graphs of Order 3p
      2
    . Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 567-575. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1751/

[1] Y.G. Baik, Y.-Q. Feng, H.S. Sim and M.Y. Xu, On the normality of Cayley graphs of abelian groups, Algebra Colloq. 5 (1998) 297-304. | Zbl 0904.05037

[2] N. Biggs, Algebraic Graph Theory, Second Ed. (Cambridge University Press, Cam- bridge, 1993). | Zbl 0284.05101

[3] W. Bosma, C. Cannon and C. Playoust, The MAGMA algebra system I: the user language, J. Symbolic Comput. 24 (1997) 235-265. doi:10.1006/jsco.1996.0125[Crossref] | Zbl 0898.68039

[4] C.Y. Chao, On the classification of symmetric graphs with a prime number of ver- tices, Trans. Amer. Math. Soc. 158 (1971) 247-256. doi:10.1090/S0002-9947-1971-0279000-7[Crossref]

[5] Y. Cheng and J. Oxley, On weakly symmetric graphs of order twice a prime, J. Combin. Theory (B) 42 (1987) 196-211. doi:10.1016/0095-8956(87)90040-2[Crossref] | Zbl 0583.05032

[6] M. Conder, Orders of symmetric cubic graphs, The Second Internetional Workshop on Group Theory and Algebraic Combinatorics, (Peking University, Beijing, 2008).

[7] M. Conder and C.E. Praeger, Remarks on path-transitivity on finite graphs, Euro- pean J. Combin. 17 (1996) 371-378. doi:10.1006/eujc.1996.0030[Crossref] | Zbl 0871.05029

[8] D.Ž. Djoković and G.L. Miller, Regular groups of automorphisms of cubic graphs, J. Combin. Theory (B) 29 (1980) 195-230. doi:10.1016/0095-8956(80)90081-7[Crossref]

[9] Y.-Q. Feng and J.H. Kwak, One-regular cubic graphs of order a small number times a prime or a prime square, J. Aust. Math. Soc. 76 (2004) 345-356. doi:10.1017/S1446788700009903[Crossref] | Zbl 1055.05078

[10] Y.-Q. Feng and J.H. Kwak, Classifying cubic symmetric graphs of order 10p or 10p2, Sci. China (A) 49 (2006) 300-319. doi:10.1007/s11425-006-0300-9[Crossref] | Zbl 1109.05051

[11] Y.-Q. Feng and J.H. Kwak, Cubic symmetric graphs of order twice an odd prime power, J. Aust. Math. Soc. 81 (2006) 153-164. doi:10.1017/S1446788700015792[Crossref]

[12] Y.-Q. Feng and J.H. Kwak, Cubic symmetric graphs of order a small number times a prime or a prime square, J. Combin. Theory (B) 97 (2007) 627-646. doi:10.1016/j.jctb.2006.11.001[Crossref] | Zbl 1118.05043

[13] Y.-Q. Feng, J.H. Kwak and K.S. Wang, Classifying cubic symmetric graphs of order 8p or 8p2, European J. Combin. 26 (2005) 1033-1052. doi:10.1016/j.ejc.2004.06.015[Crossref] | Zbl 1071.05043

[14] R. Frucht, A one-regular graph of degree three, Canad. J. Math. 4 (1952) 240-247. doi:10.4153/CJM-1952-022-9[Crossref] | Zbl 0046.40903

[15] A. Gardiner and C.E. Praeger, On 4-valent symmetric graphs, European. J. Combin. 15 (1994) 375-381. doi:10.1006/eujc.1994.1041[Crossref] | Zbl 0806.05037

[16] A. Gardiner and C.E. Praeger, A characterization of certain families of 4-valent symmetric graphs, European. J. Combin. 15 (1994) 383-397. doi:10.1006/eujc.1994.1042 | Zbl 0806.05038

[17] M. Ghasemi, A classification of tetravalent one-regular graphs of order 3p2, Colloq. Math. 128 (2012) 15-24. doi:10.4064/cm128-1-3[Crossref][WoS] | Zbl 1258.05048

[18] M. Ghasemi and J.-X. Zhou, Tetravalent s-transitive graphs of order 4p2, Graphs Combin. 29 (2013) 87-97. doi:10.007/s00373-011-1093-3 | Zbl 1258.05049

[19] D. Gorenstein, Finite Simple Groups (Plenum Press, New York, 1982). doi:10.1007/978-1-4684-8497-7[Crossref] | Zbl 0483.20008

[20] C.H. Li, Finite s-arc-transitive graphs, The Second Internetional Workshop on Group Theory and Algebraic Combinatorics, (Peking University, Beijing, 2008).

[21] C.H. Li, The finite vertex-primitive and vertex-biprimitive s-transitive graphs for s ≥ 4, Trans. Amer. Math. Soc. 353 (2001) 3511-3529. doi:10.1090/S0002-9947-01-02768-4[Crossref] | Zbl 0974.05042

[22] C.H. Li, Z.P. Lu and D. Maruˇsiˇc, On primitive permutation groups with small sub- orbits and their orbital graphs, J. Algebra 279 (2004) 749-770. doi:10.1016/j.jalgebra.2004.03.005[Crossref]

[23] C.H. Li, Z.P. Lu and H. Zhang, Tetravalent edge-transitive Cayley graphs with odd number of vertices, J. Combin. Theory (B) 96 (2006) 164-181. doi:10.1016/j.jctb.2005.07.003[Crossref] | Zbl 1078.05039

[24] R.C. Miller, The trivalent symmetric graphs of girth at most six, J. Combin. Theory (B) 10 (1971) 163-182. doi:10.1016/0095-8956(71)90075-X[Crossref]

[25] P. Potoˇcnik, P. Spiga and G. Verret. http://www.matapp.unimib.it/spiga/

[26] P. Potoˇcnik, P. Spiga and G. Verret, Cubic vertex-transitive graphs on up to 1280 vertices. arXiv:1201.5317v1 [math.CO]. | Zbl 1256.05102

[27] C.E. Praeger, An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc. 47 (1993) 227-239. doi:10.1112/jlms/s2-47.2.227[Crossref]

[28] D.J.S. Robinson, A Course in the Theory of Groups (Springer-Verlag, New York, 1982). | Zbl 0483.20001

[29] W.T. Tutte, A family of cubical graphs, Proc. Camb. Phil. Soc. 43 (1947) 459-474. doi:10.1017/S0305004100023720[Crossref] | Zbl 0029.42401

[30] R.J. Wang and M.Y. Xu, A classification of symmetric graphs of order 3p, J. Com- bin. Theory (B) 58 (1993) 197-216. doi:10.1006/jctb.1993.1037[Crossref] | Zbl 0793.05074

[31] R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica 1 (1981) 309-311. doi:10.1007/BF02579337[Crossref] | Zbl 0486.05032

[32] S. Wilson and P. Potoˇcnik, A Census of edge-transitive tetravalent graphs. http://jan. ucc.nau.edu/swilson/C4Site/index.html.

[33] M.Y. Xu, A note on one-regular graphs, Chinese Sci. Bull. 45 (2000) 2160-2162.

[34] J. Xu and M.Y. Xu, Arc-transitive Cayley graphs of valency at most four on abelian groups, Southeast Asian Bull. Math. 25 (2001) 355-363. doi:10.1007/s10012-001-0355-z [Crossref] | Zbl 0993.05086

[35] J.-X. Zhou, Tetravalent s-transitive graphs of order 4p, Discrete Math. 309 (2009) 6081-6086. doi:10.1016/j.disc.2009.05.014[Crossref][WoS]

[36] J.-X. Zhou and Y.-Q. Feng, Tetravalent s-transitive graphs of order twice a prime power, J. Aust. Math. Soc. 88 (2010) 277-288. doi:10.1017/S1446788710000066 [Crossref]