Integral Cayley graphs. (English. Russian original) Zbl 1434.05069

Guo, W.; Lytkina, D. V.; Mazurov, V. D.; Revin, D. O.

Algebra Logic 58, No. 4, 297-305 (2019); translation from Algebra Logika 58, No. 4, 445-457 (2019).

Summary: Let \(G\) be a group and \(S \subseteq G\) a subset such that \(S = S^{-1}\), where \(S^{-1} = \{s^{-1} | s \in S\}\). Then a Cayley graph \(\operatorname{Cay}(G, S)\) is an undirected graph \(\Gamma\) with vertex set \(V (\Gamma ) = G\) and edge set \(E(\Gamma ) = \{(g, gs) | g \in G, s \in S\}\). 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 \(\operatorname{Cay}(G, S)\) are integers. Using this fact, we give affirmative answers to Questions 19.50(a) and 19.50(b) in the [V. D. Mazurov (ed.) and E. I. Khukhro (ed.), The Kourovka notebook. Unsolved problems in group theory. 19th ed. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2018)].

MSC:

05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
05C50	Graphs and linear algebra (matrices, eigenvalues, etc.)

Keywords:

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

Citations:

Zbl 1211.20001

References:

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