Algebraic degree of Cayley colour graphs

The splitting field of a graph $Γ$ with respect to a square matrix $M$ associated with $Γ$, is the smallest field extension over the field of rationals $\mathbb{Q}$ that contains all the eigenvalues of $M$. The degree of the extension is called the algebraic degree of $Γ$ with respect to $M$. In this paper, we completely determine the splitting field of the adjacency matrix of the Cayley colour graph $\operatorname{Cay}(G,f)$ on a finite group $G$, associated with a class function $f:G\to\mathbb{Q}$ and compute its algebraic degree, which generalize the main results of Wu et al. Moreover, we study the relation between the algebraic integrality of two Cayley colour graphs, and deduce the fact that the algebraic degree and distance algebraic degree of a normal Cayley graph are same, generalizing a result of Zhang et al.

💡 Research Summary

The paper investigates the algebraic degree and splitting field of the adjacency matrix of Cayley colour graphs, a broad generalization of ordinary Cayley graphs. For a finite group (G) of order (n) and a class function (f\colon G\to\mathbb{Q}) satisfying (f(g)=f(g^{-1})), the authors define the undirected Cayley colour graph (\Gamma_f=\operatorname{Cay}(G,f)). Its adjacency matrix is real symmetric, so all eigenvalues are real. Using representation theory, each eigenvalue can be expressed as
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