Algebra and Logic

, Volume 58, Issue 4, pp 297–305 | Cite as

Integral Cayley Graphs

  • W. GuoEmail author
  • D. V. Lytkina
  • V. D. Mazurov
  • D. O. Revin

Let G be a group and SG a subset such that S = S−1, where S−1 = {s−1 | sS}. 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, sS}. For a normal subset S of a finite group G such that sSskS 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.


Cayley graph adjacency matrix of graph spectrum of graph integral graph complex group algebra character of group 


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  1. 1.
    Unsolved Problems in Group Theory, The Kourovka Notebook, No. 19, Institute of Mathematics SO RAN, Novosibirsk (2018); alglog/19tkt.pdf.
  2. 2.
    E. V. Konstantinova and D. V. Lytkina, “On integral Cayley graphs of finite groups,” Alg. Colloq., in print.Google Scholar
  3. 3.
    P. Diaconis and M. Shahshahani, “Generating a random permutation with random transpositions,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 57, 159-179 (1981).MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. R. Murty, Ramanujan graphs, J. Ramanujan Math. Soc., 18, No. 1, 33-52 (2003).MathSciNetzbMATHGoogle Scholar
  5. 5.
    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).MathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Chapuy and V. Féray, “A note on a Cayley graph of Sym n,” arXiv: 1202.4976v2 [math.CO].Google Scholar
  7. 7.
    I. M. Isaacs, Character Theory of Finite Groups, Corr. repr. of the 1976 orig., AMS Chelsea Publ., Providence, RI (2006).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • W. Guo
    • 1
    Email author
  • D. V. Lytkina
    • 2
    • 3
  • V. D. Mazurov
    • 4
  • D. O. Revin
    • 1
    • 3
    • 4
  1. 1.University of Science and Technology of ChinaHefeiP.R. China
  2. 2.Siberian State University of Telecommunications and Information SciencesNovosibirskRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia
  4. 4.Sobolev Institute of MathematicsNovosibirskRussia

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