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Splitting fields of mixed Cayley graphs over abelian groups

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Abstract

The splitting field \(\mathbb{S}\mathbb{F}(\Gamma )\) of a mixed graph \(\Gamma \) is the smallest field extension of \(\mathbb {Q}\) which contains all eigenvalues of the Hermitian adjacency matrix of \(\Gamma \). The extension degree \([\mathbb{S}\mathbb{F}(\Gamma ):\mathbb {Q}]\) is called the algebraic degree of \(\Gamma \). In this paper, we determine the splitting fields and algebraic degrees of mixed Cayley graphs over abelian groups. This generalizes the main results of Mönius (J Algebra 594(15):154–169, 2022). Additionally, we provide a characterization of integral mixed Cayley graphs over abelian groups, which implies the main result of Kadyan and Bhattacharjya (Electron J Combin 28(4):P4.46, 2021).

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Acknowledgements

The authors are so grateful to the referee for their valuable comments and corrections which improve the presentation of the paper. L. Lu is supported by National Natural Science Foundation of China (Grant No. 12001544) and Natural Science Foundation of Hunan Province (Grant No. 2021JJ40707). X. Huang is supported by National Natural Science Foundation of China (Grant No. 11901540).

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Correspondence to Lu Lu.

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Huang, X., Lu, L. & Mönius, K. Splitting fields of mixed Cayley graphs over abelian groups. J Algebr Comb 58, 681–693 (2023). https://doi.org/10.1007/s10801-023-01260-4

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