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 Algebra Logika, 2019, Volume 58, Number 4, Pages 445–457 (Mi al907)

Integral Cayley graphs

W. Guoa, D. V. Lytkinabc, V. D. Mazurovcd, D. O. Revincda

a Dep. Math., Univ. Sci. Tech. China, Hefei 230026, P. R. China
b Siberian State University of Telecommunications and Informatics, Novosibirsk
c Novosibirsk State University
d Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: Let $G$ be a group and $S\subseteq G$ a subset such that $S=S^{-1}$, where $S^{-1}=\{s^{-1}\mid s\in S\}$. Then the Cayley graph $\mathrm{ Cay}(G,S)$ is an undirected graph $\Gamma$ with vertex set $V(\Gamma)=G$ and edge set $E(\Gamma)=\{(g,gs)\mid g\in G, s\in S\}$. For a normal subset $S$ of a finite group $G$ such that $s\in S\Rightarrow s^k\in S$ for every $k\in \mathbb{Z}$ which is coprime to the order of $s$, we prove that all eigenvalues of the adjacency matrix of $\mathrm{ Cay}(G,S)$ are integers. Using this fact, we give affirmative answers to Questions $19.50\mathrm{ (a)}$ and $19.50\mathrm{ (b)}$ in the Kourovka Notebook.

Keywords: Cayley graph, adjacency matrix of graph, spectrum of graph, integral graph, complex group algebra, character of group.

 Funding Agency Grant Number National Natural Science Foundation of China 11771409 Siberian Branch of Russian Academy of Sciences I.1.1., ïðîåêò ¹ 0314-2016-0001 Anhui Initiative in Quantum Information Technologies AHY150200 Chinese Academy of Sciences President’s International Fellowship Initiative 2016VMA078 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.

DOI: https://doi.org/10.33048/alglog.2019.58.401

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English version:
Algebra and Logic, 2019, 58:4, 297–305

Bibliographic databases:

UDC: 512.542
Revised: 08.11.2019

Citation: W. Guo, D. V. Lytkina, V. D. Mazurov, D. O. Revin, “Integral Cayley graphs”, Algebra Logika, 58:4 (2019), 445–457; Algebra and Logic, 58:4 (2019), 297–305

Citation in format AMSBIB
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\paper Integral Cayley graphs
\jour Algebra Logika
\yr 2019
\vol 58
\issue 4
\pages 445--457
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\crossref{https://doi.org/10.33048/alglog.2019.58.401}
\transl
\jour Algebra and Logic
\yr 2019
\vol 58
\issue 4
\pages 297--305
\crossref{https://doi.org/10.1007/s10469-019-09550-2}
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V E. Konstantinova , D. Lytkina, “Integral cayley graphs over finite groups”, Algebr. Colloq., 27:1, SI (2020), 131–136
2. I. Yu. Mogil'nykh, “Perfect Codes From Pgl(2,5) in Star Graphs”, Sib. Electron. Math. Rep., 17 (2020), 534–539
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