Algebraic degree of Cayley graphs over dicyclic and semi-dihedral groups

We explore the splitting field F(X) for a simple connected graph X, which is the smallest field extension of Q that encompasses all eigenvalues of a specific adjacency matrix associated with X. The algebraic degree of X denoted as [F(X) : Q], represents the extension degree of this field. Our study focuses on deriving both upper and lower bounds for the algebraic degrees of Cayley graphs over the dicyclic group and the semi-dihedral group. Furthermore, we provide detailed analysis on the algebraic degrees and the corresponding splitting fields for normal mixed Cayley graphs over these two groups.