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On finite groups all of whose bi-Cayley graphs of bounded valency are integral | ||
| Transactions on Combinatorics | ||
| دوره 10، شماره 4، اسفند 2021، صفحه 247-252 اصل مقاله (222.78 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2021.126275.1787 | ||
| نویسنده | ||
| Majid Arezoomand* | ||
| University of Larestan, 74317-16137, Lar, Iran | ||
| چکیده | ||
| Let $k\geq 1$ be an integer and $\mathcal{I}_k$ be the set of all finite groups $G$ such that every bi-Cayley graph BCay(G,S) of $G$ with respect to subset $S$ of length $1\leq |S|\leq k$ is integral. Let $k\geq 3$. We prove that a finite group $G$ belongs to $\mathcal{I}_k$ if and only if $G\cong\Bbb Z_3$, $\Bbb Z_2^r$ for some integer $r$, or $S_3$. | ||
| کلیدواژهها | ||
| Bi-Cayley graph؛ Integer eigenvalues؛ Irreducible representation | ||
| مراجع | ||
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[1] O. Ahmadi, N. Alon, L. F. Blake and I. E. Shparlinski, Graphs with integral spectrum, Linear Algebra Appl., 430 (2009) 547–552. [2] M. Arezoomand and B. Taeri, On the characteristic polynomial of n-Cayley digraphs, Electron. J. Combin., 20 (2013) pp. 14. [3] M. Arezoomand and B. Taeri, A classification of finite groups with integral bi-Cayley graphs, Trans. Comb., 4 (2015) 55–61. [4] M. Arezoomand and B. Taeri, Finite BCI-groups are solvable, Int. J. Group Theory, 5 (2016) 1–6.
[5] M. Arezoomand and B. Taeri, Finite groups admitting a connected cubic integral bi-Cayley graph, Alg. Struc. Appl., 5 (2018) 35–43. [6] K. Baliňska, D. Cvetkovič, Z. Radosavljevič, S. Simič and D. Stevanovič, A survey on integral graphs, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat., 13 (2002) 42–65. [7] I. Estélyi and I. Kovács, On groups all of whose undirected Cayley graphs of bounded valency are integral, Electron. J. Combin., 21 no.4 (2014) pp. 11. [8] Y.-Q Feng and I. Kovács, On finite groups with prescribed two-generator subgroups and integral Cayley graphs, J. Group Theory, https://www.degruyter.com/document/doi/10.1515/jgth-2020-0094/html. [9] GAP - Groups, Algorithms, Programming, a System for Computational Discrete Algebra, Version 4.6.5, 2013, http://gap-system.org. [10] F. Harary and A. J. Schwenk, Which Graphs Have Integral Spectra?, Lecture Notes in Mathematics, 406 Springer, Berlin, (1974) 45-51. [11] Z. P. Lu, C. Q. Wang and M. Y. Xu, Semisymmetric cubic graphs constructed from bi-Cayley graphs of An, Ars Combin., 80 (2006) 177–187. [12] X. Ma and K. Wang, On finite groups all of whose cubic Cayley graphs are integral, J. Algebra Appl., 15 (2016) pp. 10. | ||
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