Splitting fields of mixed Cayley graphs over abelian groups

TL;DR

This paper determines the splitting fields and algebraic degrees of mixed Cayley graphs over abelian groups, extending previous results on circulant and integral mixed Cayley graphs.

Contribution

It generalizes existing results by explicitly characterizing the splitting fields and algebraic degrees for mixed Cayley graphs over abelian groups.

Findings

Determined the splitting fields of mixed Cayley graphs over abelian groups.

Calculated the algebraic degrees of these graphs.

Extended previous work on circulant and integral mixed Cayley graphs.

Abstract

The splitting field $\mathbb{SF}(\Gamma)$ 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 $[\mathbb{SF}(\Gamma):\mathbb{Q}]$ 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 [K. M\"{o}nius, Splitting fields of spectra of circulant graphs, J. Algebra 594(15) (2022) 154--169] and [M. Kadyan, B. Bhattacharjya, Integral mixed Cayley graphs over abelian groups, Electron. J. Combin. 28(4) (2021) \#P4.46].