Algebraic degrees of 2-Cayley digraphs over abelian groups. (English) Zbl 1540.05079
Summary: A digraph \(\Gamma\) is called a 2-Cayley digraph over a group \(G\) if there exists a 2-orbit semiregular subgroup of \(\mathrm{Aut}(\Gamma)\) isomorphic to \(G\). In this paper, we completely determine the algebraic degrees of 2-Cayley digraphs over abelian groups. This generalizes the main results of L. Lu and K. Mönius [J. Algebr. Comb. 57, No. 3, 753–761 (2023; Zbl 1514.05074)]. As applications, we consider the algebraic degrees of Cayley digraphs over finite groups admitting an abelian subgroup of index 2. Special attention is paid to the algebraic degrees of Cayley (di)graphs over generalized dihedral groups, generalized dicyclic groups and semi-dihedral groups.
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05C20 | Directed graphs (digraphs), tournaments |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 20K99 | Abelian groups |
Citations:
Zbl 1514.05074References:
| [1] | M. Arezoomand and B. Taeri, On the characteristic polynomial of n-Cayley digraphs, Electron. J. Comb. 20 (2013), research paper p57, 14, doi:10.37236/3105, https://doi.org/10. 37236/3105. · Zbl 1295.05139 · doi:10.37236/3105 |
| [2] | A. Behajaina and F. Legrand, On integral mixed cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2, 2022, arXiv:2203.08793 [math.CO]. |
| [3] | T. Cheng, L. Feng and H. Huang, Integral Cayley graphs over dicyclic group, Linear Alge-bra Appl. 566 (2019), 121-137, doi:10.1016/j.laa.2019.01.002, https://doi.org/10. 1016/j.laa.2019.01.002. · Zbl 1410.05118 · doi:10.1016/j.laa.2019.01.002 |
| [4] | T. Cheng, L. Feng, W. Liu, L. Lu and D. Stevanović, Distance powers of integral Cayley graphs over dihedral groups and dicyclic groups, Linear Multilinear Algebra 70 (2022), 1281-1290, doi:10.1080/03081087.2020.1758609, https://doi.org/10. 1080/03081087.2020.1758609. · Zbl 1487.05113 · doi:10.1080/03081087.2020.1758609 |
| [5] | X. Gao, H. Lü and Y. Hao, The Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups, J. Appl. Math. Comput. 51 (2016), 383-395, doi:10.1007/ s12190-015-0911-9, https://doi.org/10.1007/s12190-015-0911-9. · Zbl 1339.05230 · doi:10.1007/s12190-015-0911-9 |
| [6] | X. Gao and Y. Luo, The spectrum of semi-Cayley graphs over abelian groups, Linear Alge-bra Appl. 432 (2010), 2974-2983, doi:10.1016/j.laa.2009.12.040, https://doi.org/10. 1016/j.laa.2009.12.040. · Zbl 1195.05042 · doi:10.1016/j.laa.2009.12.040 |
| [7] | P. Guillot, A gentle course in local class field theory. Local number fields, Brauer groups, Ga-lois cohomology, Cambridge University Press, Cambridge, 2018, doi:10.1017/9781108377751, https://doi.org/10.1017/9781108377751. · Zbl 1423.11002 · doi:10.1017/9781108377751 |
| [8] | F. Li, Circulant digraphs integral over number fields, Discrete Math. 313 (2013), 821-823, doi: 10.1016/j.disc.2012.12.025, https://doi.org/10.1016/j.disc.2012.12.025. · Zbl 1260.05068 · doi:10.1016/j.disc.2012.12.025 |
| [9] | F. Li, A method to determine algebraically integral Cayley digraphs on finite abelian group, Contrib. Discrete Math. 15 (2020), 148-152, doi:10.11575/cdm.v15i2.62327, https:// doi.org/10.11575/cdm.v15i2.62327. · Zbl 1477.05084 · doi:10.11575/cdm.v15i2.62327 |
| [10] | X. Liu and S. Zhou, Eigenvalues of Cayley graphs, Electron. J. Comb. 29 (2022), research paper p2.9, 164, doi:10.37236/8569, https://doi.org/10.37236/8569. · Zbl 1492.05093 · doi:10.37236/8569 |
| [11] | L. Lu, Q. Huang and X. Huang, Integral Cayley graphs over dihedral groups, J. Algebr. Comb. 47 (2018), 585-601, doi:10.1007/s10801-017-0787-x, https://doi.org/10. 1007/s10801-017-0787-x. · Zbl 1394.05047 · doi:10.1007/s10801-017-0787-x |
| [12] | L. Lu and K. Mönius, Algebraic degree of Cayley graphs over abelian groups and dihedral groups, J. Algebr. Comb. (2023), doi:10.1007/s10801-022-01190-7, https://doi.org/ 10.1007/s10801-022-01190-7. · Zbl 1514.05074 · doi:10.1007/s10801-022-01190-7 |
| [13] | K. Mönius, The algebraic degree of spectra of circulant graphs, J. Number Theory 208 (2020), 295-304, doi:10.1016/j.jnt.2019.08.002, https://doi.org/10.1016/j.jnt.2019. 08.002. · Zbl 1428.05200 · doi:10.1016/j.jnt.2019.08.002 |
| [14] | K. Mönius, Splitting fields of spectra of circulant graphs, J. Algebra 594 (2022), 154-169, doi:10.1016/j.jalgebra.2021.11.036, https://doi.org/10.1016/j.jalgebra. 2021.11.036. · Zbl 1482.05203 · doi:10.1016/j.jalgebra.2021.11.036 |
| [15] | G. Sabidussi, Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426-438, http:// eudml.org/doc/177267. · Zbl 0136.44608 |
| [16] | N. Sripaisan and Y. Meemark, Algebraic degree of spectra of Cayley hypergraphs, Discrete Appl. Math. 316 (2022), 87-94, doi:10.1016/j.dam.2022.03.029, https://doi.org/10. 1016/j.dam.2022.03.029. · Zbl 1490.05171 · doi:10.1016/j.dam.2022.03.029 |
| [17] | B. Steinberg, Representation Theory of Finite Groups, Springer, New York, 2009. |
| [18] | L. L. X.Y. Huang and K. Mönius, Splitting fields of mixed cayley graphs over abelian groups, 2022, arXiv:2202.00987 [math.CO]. |
| [19] | H. Zou and J. Meng, Some algebraic properties of Bi-Cayley graphs, Acta Math. Sin., Chin. Ser. 50 (2007), 1075-1080. · Zbl 1157.05315 |
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