International Journal of Mathematical Sciences and Computing (IJMSC)
IJMSC Vol. 10, No. 1, 8 Feb. 2024
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The Construction of two classes of 4-valent tri- Cayley Graphs over Cyclic Group
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Author(s)
Xiaohan Ye
1,*
Huanzhi Zhang
1
Index Terms
tri-Cayley graph, cyclic group, vertex-transitive, automorphism group, edge-transitive, arc-transitive
Abstract
The symmetry of the graph has always been a hot topic in graph theory and the vertex-transitive graphs are a class of graphs with high symmetry. Cayley graphs which are the highly symmetrical graphs play an important role and much work has been done in the study. The tri-Cayley graph is a natural generalization of the Cayley graph. A graph is said to be a tri-Cayley graph if it admits a semiregular subgroup of automorphisms having three orbits of equal length. Koács et al. classified the cubic symmetric tricirculants in 2012 and Potočnik et al. classified the cubic vertex-transitive tricirculants in 2018. Currently, there is no research on the classification of 4-valent tri-Cayley graphs over cyclic group. In this paper, we will construct two classes of 4-valent tri-Cayley graphs over cyclic group and discuss their automorphism groups. In addition, the vertex transitivity, edge transitivity and arc transitivity are proved.
Cite This Paper
Xiaohan Ye, Huanzhi Zhang, "The Construction of two classes of 4-valent tri-Cayley Graphs over Cyclic Group", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.10, No.1, pp. 1-8, 2024. DOI: 10.5815/ijmsc.2024.01.01
Reference
[1]J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Elsevier North Holland, New York, 1976.
[2]H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
[3]K. Kutnar, D. Marušič, S. Miklavič, P. Šparl, Strongly regular tri-Cayley graphs, European Journal of Combinatorics. (2009), 30(4), 822-832.
[4]I. Koács, K. Kutnar, D. Marušič, S. Wilson, Classification of cubic symmetric tricirculants, The Electronic Journal of Combinatorics. (2012), 19(2), 24.
[5]P. Potočnik, M. Toledo, Classification of cubic vertex-transitive tricirculants, arXiv preprint arXiv: 1812.04153, 2018.
[6]D. Marušič, T. Pisanski, Symmetries of hexagonal molecular graphs on the torus, Croatica Chemica Acta. (2000), 73(4), 969-981.
[7]M. Boben, T. Pisanski, A. Žitnik, I‐graphs and the corresponding configurations, Journal of Combinatorial Designs. (2005), 13(6), 406-424.
[8]A. Devillers, M. Giudici, W. Jin, Arc-transitive bicirculants, Journal of the London Mathematical Society. (2022), 105(1), 1-23.
[9]W. Jin, Two-arc-transitive bicirculants, arXiv preprint arXiv: 2211.14520, 2022.
[10]I. Koács, B. Kuzman, A. Malnič, S. Wilson, Characterzation of edge-transitive 4-valent bicirculant, Journal of Graph Theory. (2012), 69, 441-463.
[11]T. Pisanski, A classification of cubic bicirculants, Discrete Mathematics. (2007), 307(3-5), 567-578.
[12]I. Kovács, A. Malnič, D. Marušič, Š. Miklavič, One-matching bi-Cayley graphs over abelian groups, European Journal of Combinatorics. (2009), 30(2), 602-616.
[13]J. X. Zhou, Y. Q. Feng, Cubic bi-Cayley graphs over abelian groups, European Journal of Combinatorics. (2014), 36, 679-693.
[14]M. Conder, J. X. Zhou, Y. Q. Feng, M. M. Zhang, Edge-transitive bi-Cayley graphs, Journal of Combinatorial Theory, Series B. (2020), 145, 264-306.
[15]J. M. Pan, Y. N. Zhang, An explicit characterization of cubic symmetric bi-Cayley graphs on nonabelian simple groups, Discrete Mathematics. (2022), 345(9): 112954.
[16]M. M. Zhang, J. X. Zhou, Trivalent vertex-transitive bi-dihedrants, Discrete Mathematics. (2017), 340(8), 1757-1772.
[17]M. Y. Xu, Finite group preliminary, Science Press, Bei Jing, 2014.