Geodesic
Integral Cayley graphs
Algebra i logika, Tome 58 (2019) no. 4, pp. 445-457.

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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. 
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W. Guo; D. V. Lytkina; V. D. Mazurov; D. O. Revin. Integral Cayley graphs. Algebra i logika, Tome 58 (2019) no. 4, pp. 445-457. http://geodesic.mathdoc.fr/item/AL_2019_58_4_a0/

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