The integrality of distance spectra of quasiabelian 2-Cayley graphs. (English) Zbl 1536.05243
A graph is an \(m\)-Cayley graph if its automorphism group has a semiregular subgroup isomorphic to \(G\) with \(m\) orbits. The distance spectrum of a graph is the collection of eigenvalues of its distance matrix. This paper establishes a decomposition formula for distance spectra of \(m\)-Cayley graphs, and applies it to prove several results relating to the distance matrices of certain families of \(2\)-Cayley graphs. Some of their results on \(2\)-Cayley graphs apply when the group is abelian, while others apply whenever each of the connection sets for the graph is a union of conjugacy classes. There are results on distance integrality, integrality of distance powers, distance algebraic degrees, and distance splitting fields.
Reviewer: Joy Morris (Lethbridge)
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
References:
| [1] | Alperin, R. C.; Peterson, B. L., Integral sets and Cayley graphs of finite groups. Electron. J. Comb. (2012) · Zbl 1243.05143 |
| [2] | Aouchiche, M.; Hansen, P., Distance spectra of graphs: a survey. Linear Algebra Appl., 301-384 (2014) · Zbl 1295.05093 |
| [3] | Arezoomand, M.; Taeri, B., On the characteristic polynomial of \(n\)-Cayley digraphs. Electron. J. Comb. (2013) · Zbl 1295.05139 |
| [4] | Arezoomand, M., On the Laplacian and signless Laplacian polynomials of graphs with semiregular automorphisms. J. Algebraic Comb., 21-32 (2020) · Zbl 1447.05119 |
| [5] | Babai, L., Spectra of Cayley graphs. J. Comb. Theory, Ser. B, 180-189 (1979) · Zbl 0338.05110 |
| [6] | Behajaina, A.; Legrand, F., On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2. Linear Algebra Appl., 256-273 (2023) · Zbl 1519.05113 |
| [7] | Behajaina, A.; Legrand, F., On Cayley graphs over generalized dicyclic groups. Linear Algebra Appl., 264-284 (2022) · Zbl 1498.05124 |
| [8] | Cavaleri, M.; D’Angeli, D.; Donno, A., A group representation approach to balance of gain graphs. J. Algebraic Comb., 265-293 (2021) · Zbl 1470.05071 |
| [9] | Cheng, T.; Feng, L. H.; Huang, H. L., Integral Cayley graphs over dicyclic group. Linear Algebra Appl., 121-137 (2019) · Zbl 1410.05118 |
| [10] | Cheng, T.; Feng, L. H.; Liu, W. J.; Lu, L.; Stevanović, D., Distance powers of integral Cayley graphs over dihedral groups and dicyclic groups. Linear Multilinear Algebra, 1281-1290 (2022) · Zbl 1487.05113 |
| [11] | Dalfó, C.; Fiol, M. A.; Širáň, J., The spectra of lifted digraphs. J. Algebraic Comb., 419-426 (2019) · Zbl 1429.05083 |
| [12] | Dalfó, C.; Fiol, M. A.; Pavlíková, S.; Širáň, J., Spectra and eigenspaces of arbitrary lifts of graphs. J. Algebraic Comb., 651-672 (2021) · Zbl 1477.05103 |
| [13] | Gao, X.; Lü, H. Z.; Hao, Y. F., The Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups. J. Appl. Math. Comput., 383-395 (2016) · Zbl 1339.05230 |
| [14] | Gao, X.; Luo, Y. F., The spectrum of semi-Cayley graphs over abelian groups. Linear Algebra Appl., 2974-2983 (2010) · Zbl 1195.05042 |
| [15] | Godsil, C.; Spiga, P., Rationality conditions for the eigenvalues of normal finite Cayley graphs |
| [16] | Guillot, P., A Gentle Course in Local Class Field Theory: Local Number Fields, Brauer Groups, Galois Cohomology (2018), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1423.11002 |
| [17] | Harary, F.; Schwenk, A. J., Which graphs have integral spectra? · Zbl 0305.05125 |
| [18] | Huang, J.; Li, S. C., Integral and distance integral Cayley graphs over generalized dihedral groups. J. Algebraic Comb., 921-934 (2021) · Zbl 1467.05112 |
| [19] | Huang, J.; Li, S. C., Distance-integral Cayley graphs over abelian groups and dicyclic groups. J. Algebraic Comb., 1047-1063 (2021) · Zbl 1479.05207 |
| [20] | Huang, X. Y.; Lu, L.; Mönius, K., Splitting fields of mixed Cayley graphs over abelian groups. J. Algebraic Comb., 681-693 (2023) · Zbl 1525.05078 |
| [21] | Ilić, A., Distance spectra and distance energy of integral circulant graphs. Linear Algebra Appl., 1005-1014 (2010) · Zbl 1215.05105 |
| [22] | Isaacs, I. M., Character Theory of Finite Groups (2006), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI · Zbl 1119.20005 |
| [23] | James, G.; Liebeck, M., Representations and Characters of Groups (2001), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0981.20004 |
| [24] | Klotz, W.; Sander, T., Distance powers and distance matrices of integral Cayley graphs over abelian groups. Electron. J. Comb. (2012), #p25 · Zbl 1266.05058 |
| [25] | Li, F., Circulant digraphs integral over number fields. Discrete Math., 821-823 (2013) · Zbl 1260.05068 |
| [26] | Li, F., A method to determine algebraically integral Cayley digraphs on finite abelian group. Contrib. Discrete Math., 148-152 (2020) · Zbl 1477.05084 |
| [27] | Liu, X. G.; Zhou, S. M., Eigenvalues of Cayley graphs. Electron. J. Comb., 2 (2022) · Zbl 1492.05093 |
| [28] | Lovász, L., Spectra of graphs with transitive groups. Period. Math. Hung., 191-195 (1975) · Zbl 0395.05057 |
| [29] | Lu, L.; Huang, Q. X.; Huang, X. Y., Integral Cayley graphs over dihedral groups. J. Algebraic Comb., 585-601 (2018) · Zbl 1394.05047 |
| [30] | Lu, L.; Mönius, K., Algebraic degree of Cayley graphs over abelian groups and dihedral groups. J. Algebraic Comb., 753-761 (2023) · Zbl 1514.05074 |
| [31] | Mönius, K., The algebraic degree of spectra of circulant graphs. J. Number Theory, 295-304 (2020) · Zbl 1428.05200 |
| [32] | Mönius, K., Splitting fields of spectra of circulant graphs. J. Algebra, 154-169 (2022) · Zbl 1482.05203 |
| [33] | Navarro, G., Character Theory and the McKay Conjecture (2018), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1433.20001 |
| [34] | Sripaisan, N.; Meemark, Y., Algebraic degree of spectra of Cayley hypergraphs. Discrete Appl. Math., 87-94 (2022) · Zbl 1490.05171 |
| [35] | Steinberg, B., Representation Theory of Finite Groups (2009), Springer: Springer New York |
| [36] | Terras, A., Fourier Analysis on Finite Groups and Applications (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0928.43001 |
| [37] | Wu, Y. J.; Yang, J.; Feng, L. H., Algebraic degrees of 2-Cayley digraphs over abelian groups. Ars Math. Contemp. (2024), #P2.02 · Zbl 1540.05079 |
| [38] | Wu, Y. J.; Zhang, X. Q.; Feng, L. H.; Wu, T. Z., Distance and adjacency spectra and eigenspaces for three (di)graph lifts: a unified approach. Linear Algebra Appl., 147-181 (2023) · Zbl 1517.05111 |
| [39] | Zou, H.; Meng, J. X., Some algebraic properties of bi-Cayley graphs. Acta Math. Sinica (Chin. Ser.), 1075-1080 (2007), (in Chinese) · Zbl 1157.05315 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
