International Journal of Mathematics Trends and Technology
Volume 68 | Issue 10 | Year 2022 | Article Id. IJMTT-V68I10P505 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I10P505
| Received | Revised | Accepted | Published |
|---|---|---|---|
| 30 Aug 2022 | 08 Oct 2022 | 18 Oct 2022 | 31 Oct 2022 |
The vertex-transitive graph is a graph with high symmetry. A graph Γ is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two orbits of equal size. And Γ is normal with respect to H if R(H) is normal subgroup of Aut(Γ). In this paper, we complete the classification of the cubic vertex-transitive normal bi-Cayley graphs over a group of order pq2, where p and q be two primes with p>q. Furthermore, these cubic vertex-transitive bi-Cayley graphs are also a Cayley graph.
[1] M. J. D. Resmini, D. Jungnickel, “Strongly Regular Semi-Cayley Graphs,” Journal of Algebraic Combinatorics., vol. 1, no. 2, pp. 171-195, 1992.
[2] D. Marušič, T. Pisanski, “Symmetries of hexagonal molecular graphs on the torus,” Croatica Chemica Acta., vol. 73, no. 4, pp. 969- 981, 2000
[3] T. Pisanski, “A classification of cubic bicirculants,” Discrete Mathematics., vol. 307, no. (3-5), pp. 567-578, 2007.
[4] I. Kovács, B. Kuzman, A. Malnič and S. Wilson, “Characterzation of edge-transitive 4-valent bicirculant,” Journal of Graph Theory., vol. 69, pp. 441-463, 2012.
[5] I. Kovács. A. Malnič, D. Marušič and Š. Miklavič, “One-matching bi-Cayley graphs over abelian groups,” European Journal of Combinatorics., vol. 30, no. 2, pp. 602-616, 2009.
[6] J. X. Zhou, Y. Q. Feng, “Cubic bi-Cayley graphs over abelian groups,” European Journal of Combinatorics., vol. 36, pp. 679-693, 2014.
[7] A. Araluze, I. Kovács, et al, “Partial sum quadruples and bi-Abelian digraphs,” Journal of Combinatorial, Theory Series A., vol. 119, no. 8, pp. 1811-183, 2012.
[8] H. Koike, I. Koávcs, “Arc-transitive cubic abelian Bi-Cayley graphs and BCI- graphs,” Filomat., vol. 30, no. 2, pp. 321-331, 2016.
[9] J. M. Pan, Y. N. Zhang, “An explicit characterization of cubic symmetric bi-Cayley graphs on nonabelian simple groups,” Discrete Mathematics., vol. 345, no. 9, pp. 112954, 2022.
[10] M. M. Zhang, J. X. Zhou, “Trivalent vertex-transitive bi-dihedrants,” Discrete Mathematics., vol. 340, no. 8, pp. 1757-1772, 2017.
[11] W. Xue, “Cubic Edge-Transitive bi-Cayley Graphs on Generalized Dihedral Group,” Bulletin of the Malaysian Mathematical Sciences Society., vol. 45, no. 1, pp. 1-11, 2021.
[12] M. Conder. J. X. Zhou, Y.Q. Feng and M.M. Zhang, “Edge-transitive bi-Cayley graphs,” Journal of Combinatorial Theory, Series B., vol. 145, pp. 264-306, 2020.
[13] M. Arezoomand, A. Behmaram. M. Ghasemi and P. Raeighasht, “Isomorphisms of bi-Cayley graphs on Dihedral groups,” Discrete Mathematics, Algorithms and Applications., vol. 12, no. 4, pp. 9, 2020.
[14] Y. L. Qin, J. X. Zhou, “Cubic edge-transitive bi-Cayley graphs over inner-abelian" p-groups,” Communications in Algebra., vol. 47, no. 5, pp. 1973-1984, 2019.
[15] J. X. Zhou, Y. Q. Feng, “Cubic Non-Cayley vertex-transitive bi-Cayley graphs over a regular p-group," The Electronic Journal of Combinatorics., vol. 23, no. 3, pp. P3.34, 2016.
[16] W. Jin, W. J. Liu, “A classification of nonabelian simple 3-BCI-groups,” European Journal Combinatorics., vol. 31, no. 12, pp. 1257-126, 2010.
[17] K. Kutnar, D. Marušič and C. Zhang, “On cubic non-Cayley vertex-transitive graphs,” Journal Graph Theory., vol. 69, pp. 77-95, 2012.
[18] J. A. Davis, L. Martínez and M. J. Sodupe, “Bi-Cayley normal uniform multiplicative designs,” Discrete Mathematics., vol. 339, no. 9, pp. 2224-2230, 2016.
[19] M. Arezoomand, B. Taeri, “Normality of 2-Cayley digraphs,” Discrete Mathematics., vol. 338, no. 3, pp. 41-47, 2014.
[20] A. Hujdurović, K. Kutnar and D. Marušič, “On normality of n-Cayley graphs,” Applied Mathematics and Computation., vol. 332, pp. 469-476, 2018.
[21] J. X. Zhou, “Every finite group has a normal bi-Cayley graph,” Ars Mathematica Contemporanea., vol. 14, no. 1, pp. 177-186, 2018.
[22] J. X. Zhou, Y. Q. Feng, “The automorphism of bi-Cayley graphs,” Journal of Combinatorial Theory, Series B., vol. 116, pp. 504- 532, 2016.
[23] M. Y. Xu, Finite Groups preliminary, 1nd ed., L. Yang, Ed. BeiJing: Science press, 1999.
[24] N. L. Biggs, Algebraic Graph Theory, 2nd ed., NewYork: Cambridge University press, 1993.
[25] L. C. Zhang, G. Y. Chen and S. M. Chen, “The automorphism groups of certain groups of order pq2," Journal of Southwest China Normal University (Natural Science)., vol. 28, no. 1, pp. 15-17, 2003.
Weihua Yang, "Cubic Vertex-Transitive bi-Cayley Graphs over a Nonabelian Group," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 10, pp. 28-35, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I10P505
PDF 
Abstract 
Keywords 
References 
Citation