Cubic semisymmetric graphs of order $44p$ or $44p^{2}$

Document Type : Research Paper

Authors

Faculty of Mathematical and Computer Science, kharazmi university, Tehran, Iran

Abstract

A simple graph is called semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let $p$ be an arbitrary prime. Folkman [J. Folkman, Regular line-symmetric graphs, J. Combinatorial Theory, \textbf{3} (1967) 215--232.] proved that there are no cubic semisymmetric graphs of order $2p$ or $2p^{2}$. In this paper, an extension of his result in the case of cubic graphs of order $44p$ or $44p^{2}$ is given. By using group theoretic methods, we prove that there are no connected cubic semisymmetric graphs of order $44p$ or $44p^{2}$.

Keywords

Main Subjects


[1] M. Alaeiyan and M. Lashani, A Classification of semisymmetric cubic graphs of order 28p2 , J. Indones. Math. Soc., 16 (2010) 139–143.
[2] M. Alaeiyan and B. N. Onagh, Cubic edge-transitive graphs of order 4p2 , Acta Math. Univ. Comenian. (N.S.), 78 (2009) 183–186.
[3] M. Alaeiyan and B. N. Onagh, On semisymmetric cubic graphs of order 10p3 , Hacet. J. Math. Stat., 40 (2011) 40 531–535.
[4] I. Z. Bouwer, An edge but not vertex transitive cubic graph, Canad. Math. Bull., 11 (1968) 533–535.
[5] Y. Bugeand, Z. Cao and M. Mignoto, On simple K4 -groups, J. Algebra, 241 (2001) 658–668.
[6] M. Conder, M. Malnic, D. Marusic and P. Potocnic, A census of semisymmetric cubic graphs on up to 768 vertices, J. Algebraic Combin., 23 (2006) 255–294.
[7] M. Conder and R. Nedela, A refined classification of symmetric cubic graphs, J. Algebra, 322 (2009) 722–740.
[8] M. R. Darafsheh and M. Shahsavarn, Semisymmetric cubic graphs of order 34p3 , Bull. Korean Math. Soc., 57 (2020) 739–750.
[9] S.-F. Du and D. Marusic, An infinite family of biprimitive semisymmetric graphs, J. Graph Theory, 32 (1999) 217–228.
[10] S.-F. Du and M. Xu, A classification of semisymmetric graphs of order 2pq, Comm. Algebra, 28 (2000) 2685–2715.
[11] Y. Feng, M. Ghasemi and W. Changqun, Cubic semisymmtric grphs of order 6p3 , Discrete Math., (2010) 2345–2355.
[12] Y. Feng, J. Kwak and K. Wang, Classifying cubic symmetric graphs of order 8p or 8p2 , European J. Combin., 26 (2005) 1033–1052.
[13] J. Folkman, Regular line-symmetric graphs, J. Combinatorial Theory, 3 (1967) 215–232.
[14] D. Goldschmidt, Automorphisms of trivalent graphs, Ann. of Math. (2), 111 (1980) 377–406.
[15] H. Han and Z. Lu, Semisymmetric graphs of order 6p2 and prime valency, Sci. China Math., 55 (2011) 2579–2592.
[16] M. Herzog, On finite simple groups of order divisible by three primes only, J. Algebra, 10 (1968) 383–388.
[17] X. Hua and Y. Feng, Cubic semisymmetric graphs of order 8p3 , Sci. China Math., 54 (2011) 1937–1949.
[18] M. E. Iofinova and A. A. Ivanov, Biprimitive cubic graphs, (Russian), Investigations in the algebraic theory of combinatorial objects (Russian), Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, 1985 123–134.
[19] M. H. Klin, On edge but not vertex transitive graphs, Algebraic methods in graph theory, I, II, (1978), 399–403, Colloq. Math. Soc. János Bolyai, 25, North-Holland, Amsterdam-New York, 1981.
[20] Z. Lu, C. Wang and M. Xu, On semisymmetric cubic graphs of order 6p2 , Sci. China Ser. A, 47 (2004) 1–17.
[21] A. Malnic, D. Marusic and C. Wang, Cubic semisymmetric graphs of order 2p3 , Ljubljana, Slovenia: University of Ljubljana Preprint Series, 38 2000.
[22] A. Malnic, D. Marusic and C. Wang, Cubic edge-transitive graphs of order 2p3 , Discrete Math., 274 (2004) 187–198.
[23] C. Parker and P. Rowley, Classical groups in dimension 3 as completions of the Goldschmidt G3 -amalgam, J. London Math. Soc. (2), 62 (2000) 802–812.
[24] D. J. S. Robinson, A course in the theory of groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York-Berlin, 1982.
[25] M. Shahsavaran and M. R Darafsheh, On semisymmetric cubic graphs of order 20p2 , p prime, Discuss. Math. Graph Theory, (2021) 873–891.
[26] M. Shahsavaran and M. R. Darafsheh, Classifying semisymmetric cubic graphs of order 20p, Turkish J. Math., 43 (2019) 2755–2766.
[27] Y. Bugeaud, Z. Cao, M. Mignotte, On simple K4 -groups, J. Algebra, 241 (2001) 658–668.
[28] M. Suzuki, Group theory, II, Translated from the Japanese, Grundlehren der mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences], 248, Springer-Verlag, New York, 1986.
[29] A. Talebi and N. Mehdipoor, Classifying cubic semisymmetric graphs of order 18pn , Graphs Combin., 30 (2014) 1037–1044.
[30] W. T. Tutte, Connectivity in graphs, Mathematical Expositions, University of Toronto Press, Toronto, Ont.; Oxford University Press, London 1966.
[31] C. Q. Wang and T. S Chen, Semisymmetric cubic graphs as regular covers of K3,3 , Acta Math. Sin. (Engl. Ser.), 24 (2008) 405–416.
[32] S. Zhang and W. J. Shi, Revisiting the number of simple K4 -groups, arXiv:1307.8079v1 [math.NT] 2013 pp 9.
  • Receive Date: 20 February 2023
  • Revise Date: 28 July 2023
  • Accept Date: 12 August 2023
  • Published Online: 01 June 2024