An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. In this paper, we classify all connected cubic symmetric graphs of order 52p2 for each prime p.
1. Alaeiyan, M. and Hosseinipoor M. K. A classification of the cubic s-regular graphs of orders 12p and 12p2. Acta Univ. Apulensis Math. Inform., 2011, 25, 153–158.
2. Cheng, Y. and Oxley, J. On weakly symmetric graphs of order twice a prime. J. Combin. Theory Ser. B, 1987, 42(2), 196–211. 3. Conder, M. Trivalent (Cubic) Symmetric Graphs on up to 10000 Vertices. 2011.
https://www.math.auckland.ac.nz/~conder/symmcubic10000list.txt
https://doi.org/10.1016/0095-8956(87)90040-2
4. Conway, J. H., Curties, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A. Atlas of Finite Groups. Clarendon Press, Oxford, 1985.
5. Feng, Y. Q., Ghasemi, M. and Yang, D. W. Cubic symmetric graphs of order 8p3. Discrete Math., 2014, 318, 62–70.
https://doi.org/10.1016/j.disc.2013.11.013
6. Feng, Y. Q. and Kwak, J. H. One-regular cubic graphs of order a small number times a prime or a prime square. J. Aust. Math. Soc., 2004, 76(3), 345–356.
https://doi.org/10.1017/S1446788700009903
7. Feng, Y. Q. and Kwak, J. H. Classifying cubic symmetric graphs of order 10p or 10p2. Sci. China Ser. A, 2006, 49(3), 300–319.
https://doi.org/10.1007/s11425-006-0300-9
8. Feng, Y. Q. and Kwak, J. H. Cubic symmetric graphs of order twice an odd prime-power. J. Aust. Math. Soc., 2006, 81(2), 153–164.
https://doi.org/10.1017/S1446788700015792
9. Feng, Y. Q. and Kwak, J. H. Cubic symmetric graphs of order a small number times a prime or a prime square. J. Combin. Theory Ser. B, 2007, 97(4), 627–646.
https://doi.org/10.1016/j.jctb.2006.11.001
10. Feng, Y. Q., Kwak, J. H. and Wang, K. Classifying cubic symmetric graphs of order 8p or 8p2, European J. Combin., 2005, 26(7), 1033–1052.
https://doi.org/10.1016/j.ejc.2004.06.015
11. Feng, Y. Q., Kwak, J. H. and Xu, M. Y. Cubic s-regular graphs of order 2p3. J. Graph Theory, 2006, 52(4), 341–352.
https://doi.org/10.1002/jgt.20169
12. Gorenstein, D., Lyons, R. and Solomon, R. The Classification of the Finite Simple Groups. Mathematical Surveys and Monographs, Vol. 40, American Mathematical Society, Providence, RI, 1994.
https://doi.org/10.1090/surv/040.1
13. Kelarev, A. V. Graph Algebras and Automata. Marcel Dekker, New York, 2003.
https://doi.org/10.1201/9781482276367
14. Lorimer, P. Vertex-transitive graphs: Symmetric graphs of prime valency. J. Graph Theory, 1984, 8(1), 55–68.
https://doi.org/10.1002/jgt.3190080107
15. Oh, J. M. Cubic s-regular graphs of order 12p, 36p, 44p, 52p, 66p, 68p and 76p. J. Appl. Math. Inform., 2013, 31(5–6), 651–659.
https://doi.org/10.14317/jami.2013.651
16. Rotman, J. J. An Introduction to the Theory of Groups. Springer, New York, 1995.
https://doi.org/10.1007/978-1-4612-4176-8
17. Talebi, A. A. and Mehdipoor, N. Classifying cubic s-regular graphs of orders 22p and 22p2. Algebra Discrete Math., 2013, 16(2), 293–298.
18. Tutte, W. T. A family of cubical graphs. Proc. Cambridge Philos. Soc., 1947, 43, 459–474.
https://doi.org/10.1017/S0305004100023720
19. Tutte, W. T. On the symmetry of cubic graphs. Canadian J. Math., 1959, 11, 621–624.
https://doi.org/10.4153/CJM-1959-057-2
20. Zhang, M. Software defined network energy efficient algorithm based on degree sequence of nodes. Microelectr. Comp., 2021, 38(10), 65–72.
21. Zhou, J. X. and Feng, Y. Q. Cubic vertex-transitive graphs of order 2pq. J. Graph Theory, 2010, 65(4), 285–302.
https://doi.org/10.1002/jgt.20481