Let G be a group and S ⊆ G a subset such that S = S−1, where S−1 = {s−1 | s ∈ S}. Then a Cayley graph Cay(G, S) is an undirected graph Γ with vertex set V (Γ) = G and edge set E(Γ) = {(g, gs) | g ∈ G, s ∈ S}. For a normal subset S of a finite group G such that s ∈ S ⇒ sk ∈ S for every k ∈ ℤ which is coprime to the order of s, we prove that all eigenvalues of the adjacency matrix of Cay(G, S) are integers. Using this fact, we give affirmative answers to Questions 19.50(a) and 19.50(b) in the Kourovka Notebook.
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Unsolved Problems in Group Theory, The Kourovka Notebook, No. 19, Institute of Mathematics SO RAN, Novosibirsk (2018); http://math.nsc.ru/ alglog/19tkt.pdf.
E. V. Konstantinova and D. V. Lytkina, “On integral Cayley graphs of finite groups,” Alg. Colloq., in print.
P. Diaconis and M. Shahshahani, “Generating a random permutation with random transpositions,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 57, 159-179 (1981).
M. R. Murty, Ramanujan graphs, J. Ramanujan Math. Soc., 18, No. 1, 33-52 (2003).
R. Krakovski and B. Mohar, “Spectrum of Cayley graphs on the symmetric group generated by transpositions,” Lin. Alg. Appl., 437, No. 3, 1033-1039 (2012).
G. Chapuy and V. Féray, “A note on a Cayley graph of Symn,” arXiv: 1202.4976v2 [math.CO].
I. M. Isaacs, Character Theory of Finite Groups, Corr. repr. of the 1976 orig., AMS Chelsea Publ., Providence, RI (2006).
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W. Guo Supported by the NNSF of China (grant No. 11771409) and by Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences and Anhui Initiative in Quantum Information Technologies (grant No. AHY150200).
V. D. Mazurov Supported by SB RAS Fundamental Research Program I.1.1, project No. 0314-2016-0001.
D. O. Revin Supported by Chinese Academy of Sciences President’s International Fellowship Initiative, grant No. 2016VMA078.
Translated from Algebra i Logika, Vol. 58, No. 4, pp. 445-457, July-August, 2019.
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Guo, W., Lytkina, D.V., Mazurov, V.D. et al. Integral Cayley Graphs. Algebra Logic 58, 297–305 (2019). https://doi.org/10.1007/s10469-019-09550-2
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DOI: https://doi.org/10.1007/s10469-019-09550-2