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Hua, Xiao-Hui; Xu, Shang-Jin; Deng, Yun-Ping

Tetravalent edge-transitive Cayley graphs of \({\mathrm{PGL}(2,p)}\). (English) Zbl 1283.05125

Acta Math. Appl. Sin., Engl. Ser. 29, No. 4, 837-842 (2013).
Summary: Let \(\varGamma=\mathrm{Cay}(G,S)\), \(R(G)\) be the right regular representation of \(G\). The graph \(\varGamma\) is called normal with respect to \(G\), if \(R(G)\) is normal in the full automorphism group \(\mathrm{Aut}(\varGamma)\) of \(\varGamma\). \(\varGamma\) is called a bi-normal with respect to \(G\) if \(R(G)\) is not normal in \(\mathrm{Aut}(\varGamma)\), but \(R(G)\) contains a subgroup of index 2 which is normal in \(\mathrm{Aut}(\varGamma)\). In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on \(\mathrm{PGL}(2,p)\) are either normal or bi-normal when \(p\neq 11\) is a prime.

Cited in 4 Documents

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C62 Graph representations (geometric and intersection representations, etc.)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures

Keywords:

Cayley graph; normal; bi-normal; simple group

Software:

Magma
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Full Text: DOI

References:

[1] Baik, Y. G., Feng, Y.Q., Sim, H.S. The normality of Cayley graphs of finite Abelian groups with valency 5. Systems Sci. Math. Sci., 13: 425–431 (2000) · Zbl 0973.05038
[2] Baik, Y. G., Feng, Y.Q., Sim, H.S., Xu, M.Y. On the normality of Cayley graphs of Abelian groups. Algebra Colloq., 5: 297–304 (1998) · Zbl 0904.05037
[3] Biggs, N. Algebraic Graph theory (Second edition). Cambridge university Press, Cambridge, 1993 · Zbl 0805.68094
[4] Bosma, W., Cannon, C., Playoust, C. The MAGMA algebra system I: The user language. J. Symbolic Comput., 24: 235–265 (1997) · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125
[5] Cameron, P. J., Omidi, G.R., Tayfeh-Rezaie, B. 3-Designs from PGL(2, q). Electron. J. Combin., 13: 1–11 (2006) · Zbl 1097.05007
[6] Chao, C. Y. On the classification of symmetric graphs with a prime number of vertices. Trans. Amer. Math. Soc., 158: 247–256 (1971) · Zbl 0217.02403 · doi:10.1090/S0002-9947-1971-0279000-7
[7] Conder, M. D.E., Li, C.H., Praeger, C.E. On the Weiss conjucture for finite locally primitive graphs. Pro. Edinbrugn Math. Soc., 43: 129–138 (2000) · Zbl 0949.05037 · doi:10.1017/S0013091500020745
[8] Conway, J. H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A. Atlas of finite groups. Clarendon Press, Oxford, 1985 · Zbl 0568.20001
[9] Dobson, E., Witte, D. Transitive permutation groups of prime-squared degree. J. Algebraic Combin., 16: 43–69 (2002) · Zbl 1027.20001 · doi:10.1023/A:1020882414534
[10] Du, S. F., Wang, R.J., Xu, M.Y. On the normality of Cayley digraphs of order twice a prime. Australasian Journal of Combinatorics, 18: 227–234 (1998) · Zbl 0917.05037
[11] Fang, X. G., Li, C.H., Xu, M.Y. On edge-transitive Cayley graphs of valency four. European J. Combin., 25: 1107–1116 (2004) · Zbl 1050.05063 · doi:10.1016/j.ejc.2003.07.008
[12] Fang, X. G., Li, C.H., Wang, D.J., Xu, M.Y. On cubic Cayley graphs of finite simple groups. Discrete Math., 244: 67–75 (2002) · Zbl 1007.05060 · doi:10.1016/S0012-365X(01)00075-9
[13] Feng, Y. Q., Xu, M.Y. Automorphism groups of tetravalent Cayley graphs on regular p-groups. Discrete Math., 305: 354–360 (2005) · Zbl 1080.05042 · doi:10.1016/j.disc.2005.10.009
[14] Feng, Y. Q., Kwak, J.H., Wang, R.J. Automorphism groups of 4-valent connected Cayley graphs of p-groups. Chinese Ann. Math. Ser. B, 22: 281–286 (2001) · Zbl 0983.05049 · doi:10.1142/S0252959901000292
[15] Feng, Y. Q., Wang, D.J., Chen, J.L. A family of nonnormal Cayley digraphs. Acta Math. Sinica (N.S.), 17: 147–152 (2001) · Zbl 0980.05030 · doi:10.1007/s101140000097
[16] Feng, Y. Q., Wang, R.J., Xu, M.Y. Automorphism groups of 2-valent connected Cayley digraphs on regular p-groups. Graphs Combin., 18: 253–257 (2002) · Zbl 0991.05055 · doi:10.1007/s003730200018
[17] Godsil, C. D. On the full automorphism group of a graph. Combinatorica, 1: 243–256 (1981) · Zbl 0489.05028 · doi:10.1007/BF02579330
[18] Guralnick, R. Subgroups of prime power index in a simple group. J. Algebra., 81: 304–311 (1983) · Zbl 0515.20011 · doi:10.1016/0021-8693(83)90190-4
[19] Klendman, P. B., Liebeck, M.W. The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Notes. Cambridge University Press, New York, Sydney, 1990
[20] Klin, R. H., Pöschel, R. The König problem, the isomorphism problem for circulant graphs and the method of Schur. Proceedings of the International Colloquium on Algebraic Methods in Graph Theory, Szeged, 1978, Coll. Mat. Soc. János Bolyai 27 · Zbl 0416.05044
[21] Li, C. H. On isomoporphisms of connected Cayley graphs III. Bull, Austral. Math. Soc., 58: 137–145 (1998) · Zbl 0908.05047 · doi:10.1017/S0004972700032068
[22] Li, C. H. Finite s-arc transitive Cayley graphs and flag-transitive projective planes. Proc. Amer. Math. Soc., 133: 31–41 (2005) · Zbl 1097.05021 · doi:10.1090/S0002-9939-04-07549-5
[23] Li, C. H. Finite edge-transitive Cayley graphs and rotary Cayley maps. Trans. Amer. Math. Soc., 358: 4605–4635 (2006) · Zbl 1112.05051 · doi:10.1090/S0002-9947-06-03900-6
[24] Li, C. H., Lu, Z.P., Zhang, H. Tetravalent edge-transitive Cayley graphs with odd number of vertices. J. Combin. Theroy B, 96: 164–181 (2006) · Zbl 1078.05039 · doi:10.1016/j.jctb.2005.07.003
[25] Li, C. H., Pan, J.M. Finite 2-arc-transitive abelian Cayley graphs. European J. Combin., 29: 148–158 (2008) · Zbl 1193.05089 · doi:10.1016/j.ejc.2006.12.001
[26] Pan, J. M. Locally primitive normal Cayley graphs of metacyclic groups. The electronic Journal of Combinatorics, 16: #R96 (2009) · Zbl 1186.05068
[27] Praeger, C. E. Finite normal edge-transitive Cayley graphs. Bull. Austral. Math. Soc., 60: 207–220 (1999) · Zbl 0939.05047 · doi:10.1017/S0004972700036340
[28] Wang, C. Q., Wang, D.J., Xu, M.Y. On normal Cayley graphs of finite groups. Science in China Ser. A, 28: 131–139 (1998) · Zbl 0912.05042
[29] Wang, C. Q., Xu, M.Y. Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D 2n . European J. Combin., 27: 750–768 (2006) · Zbl 1176.05037 · doi:10.1016/j.ejc.2004.12.007
[30] Weiss, R. The nonexistence of 8-transitive graphs. Combinatorica, 1 309–311 (1981) · Zbl 0486.05032 · doi:10.1007/BF02579337
[31] Wilson, S., Potoñik, P. A census of edge-transitive tetravalent graphs: Mine-Census. Avaliable at http://jan.ucc.nau.edu/swilson/C4Site/index.html
[32] Xu, M. Y. Automorphism groups and isomorphisms of Cayley digraphs. Discrete Math., 182: 309–319 (1998) · Zbl 0887.05025 · doi:10.1016/S0012-365X(97)00152-0
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