Algebra and Logic

, Volume 58, Issue 4, pp 297–305

# Integral Cayley Graphs

Article

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.

## Keywords

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

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## 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