Classifying cubic symmetric graphs of order $18 p^2$

Authors

  • Mehdi Alaeiyan Iran University of Science and Technology
  • Mohammad Kazem Hosseinipoor Iran University of Science and Technology
  • Masoumeh Akbarizadeh Iran University of Science and Technology

DOI:

https://doi.org/10.52737/18291163-2020.12.1-1-11

Keywords:

Symmetric graphs, $s$-regular graphs regular coverings

Abstract

A $s$-arc in a graph is an ordered $(s+1)$-tuple $(v_{0}, v_{1}, \cdots, v_{s-1}, v_{s})$ of vertices such that $v_{i-1}$ is adjacent to $v_{i}$ for $1\leq i \leq s$ and $v_{i-1}\neq v_{i+1}$ for $1\leq i < s$. A graph $X$ is called $s$-regular if its automorphism group acts regularly on the set of its $s$-arcs. In this paper, we classify all connected cubic $s$-regular graphs of order $18p^2$ for each $s\geq1$ and each prime $p$.

References

M. Alaeiyan and M. Hosseinipoor, A classification of the cubic s-regular graphs of orders $12p$ and $12p^{2}$, Acta Universitatis Apulensis, (2011), no. 25, p. 153-158.

M. Alaeiyan and M. Hosseinipoor, Cubics symmetric graphs of order $6p^{3}$, Matematicki Vesnik, 69 (2017), no. 2, pp. 101-117.

M. Alaeiyan, B. Onagh, and M. Hosseinipoor, A classification of cubic symmetric graphs of order $16p^{2}$, Proceedings-Mathematical Sciences, 121 (2011), no. 3, pp. 249-257. https://doi.org/10.1007/s12044-011-0029-4

M. Alaeiyan, L. Pourmokhtar, and M. Hosseinipoor, Cubic symmetric graphs of orders $36 p$ and $36 p^{2}$, Journal of Algebra and Related Topics, 2 (2014), no. 1, pp. 55-63.

N. Biggs, Algebraic Graph Theory, Cambridge University Press, 1974, 1993.

M. Conder and P. Dobcsanyi, Trivalent symmetric graphs on up to 768 vertices, in J. Combin. Math. Combin. Comput, Citeseer, 2002.

M. D. Conder and Y.-Q. Feng, Arc-regular cubic graphs of order four times an odd integer, Journal of Algebraic Combinatorics, 36 (2012), no. 1, p. 21-31. https://doi.org/10.1007/s10801-011-0321-5

M. D. Conder and C. E. Praeger, Remarks on path-transitivity in finite graphs, European Journal of Combinatorics, 17 (1996), no. 4, p. 371-378. https://doi.org/10.1006/eujc.1996.0030

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, 1985.

D.Ž. Djoković, and G. L. Miller, Regular groups of automorphisms of cubic graphs, Journal of Combinatorial Theory, Series B, 29 (1980), no. 2, p. 195-230. https://doi.org/10.1016/0095-8956(80)90081-7

S.-F. Du, Y.-Q. Feng, J. H. Kwak, and M.-Y. Xu, Cubic Cayley graphs on dihedral groups, Mathematical Analysis and Applications, 7 (2004), p. 224-234.

Y. Feng and J. H. Kwak, Classifying cubic symmetric graphs of order $10p$ or $10p^{ 2}$, Science in China Series A, 49 (2006), no. 3, p. 300-319. https://doi.org/10.1007/s11425-006-0300-9

Y.-Q. Feng and J. H. Kwak, $s$-Regular cubic graphs as coverings of the complete bipartite graph $K_{3, 3}$, Journal of Graph Theory, 45 (2004), no. 2, p. 101-112. https://doi.org/10.1002/jgt.10151

Y.-Q. Feng and J. H. Kwak, Cubic symmetric graphs of order twice an odd prime-power, Journal of the Australian Mathematical Society, 81 (2006), no. 2, p. 153-164. https://doi.org/10.1017/s1446788700015792

Y.-Q. Feng, and J. H. Kwak, Cubic symmetric graphs of order a small number times a prime or a prime square, Journal of Combinatorial Theory, Series B, 97 (2007), no. 4, p. 627-646. https://doi.org/10.1016/j.jctb.2006.11.001

Y.-Q. Feng, J. H. Kwak, and K. Wang, Classifying cubic symmetric graphs of order $8p$ or $8p^{2}$,

European Journal of Combinatorics, 26 (2005), no. 7, p. 1033-1052.

Y.-Q. Feng, J. H. Kwak, and M.-Y. Xu, Cubic s-regular graphs of order $2p^{3}$, Journal of Graph Theory, 52 (2006), no. 4, p. 341-352. https://doi.org/10.1002/jgt.20169

D. Gorenstein, Finite Simple Groups, University Series in Mathematics, 1982.

J. L. Gross, and T. W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Mathematics, 18 (1977), no. 3, p. 273-283. https://doi.org/10.1016/0012-365x(77)90131-5

A. Imani, N. Mehdipoor, and A. A. Talebi, On application of linear algebra in classification cubic s-regular graphs of order $28p$, Algebra and Discrete Mathematics, 25 (2018), no. 1, p. 56-72.

P. Lorimer, Vertex-transitive graphs: Symmetric graphs of prime valency, Journal of graph theory, 8 (1984), no. 1, p. 55-68. https://doi.org/10.1002/jgt.3190080107

D. Marusic, and T. Pisanski, Symmetries of hexagonal molecular graphs on the torus, Croatica Chemica Acta, 73 (2000), no. 4, p. 969-981.

J.-M. Oh, Arc-transitive elementary abelian covers of the Pappus graph, Discrete Mathematics, 309 (2009), p. 6590-6611. https://doi.org/10.1016/j.disc.2009.07.010

J.-M. Oh, A classification of cubic s-regular graphs of order $14p$, Discrete Mathematics, 309 (2009), no. 9, p. 2721-2726.

J.-M. Oh, A classification of cubic s-regular graphs of order $16p$, Discrete mathematics, 309 (2009), no. 10, p. 3150-3155.

D. Robinson, A Course in the Theory of Groups, 1982.

A. A. Talebi and N. Mehdipoor, Classifying cubic s-regular graphs of orders $22p$ and 22p$^2$, Algebra and Discrete Mathematics, (2013).

W. T. Tutte, A family of cubical graphs, Mathematical Proceedings of the Cambridge Philosophical Society, 43 (1947), no. 4, p. 459-474. https://doi.org/10.1017/s0305004100023720

W. T. Tutte, On the symmetry of cubic graphs, Canadian Journal of Mathematics, 11 (1959), p. 621-624.

H. Wielandt, Finite permutation groups, Academic Press, 2014. https://doi.org/10.4153/cjm-1959-057-2

M.-Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Mathematics, 182 (1998), p. 309-319. https://doi.org/10.1016/s0012-365x(97)00152-0

Downloads

Published

2020-03-28 — Updated on 2022-08-28

Versions

How to Cite

Classifying cubic symmetric graphs of order $18 p^2$. (2022). Armenian Journal of Mathematics, 12(1), 1-11. https://doi.org/10.52737/18291163-2020.12.1-1-11 (Original work published 2020)