H-integral and Gaussian integral normal mixed Cayley graphs

TL;DR

This paper characterizes when normal mixed Cayley graphs have integer or Gaussian integer eigenvalues, establishing an equivalence between H-integrality and Gaussian integrality for these graphs.

Contribution

It provides a complete characterization of the sets S for which normal mixed Cayley graphs are H-integral and proves their equivalence to Gaussian integrality.

Findings

Characterization of sets S for H-integrality in normal mixed Cayley graphs

Proof that H-integral and Gaussian integral properties are equivalent in these graphs

Extension of eigenvalue integrality concepts to mixed Cayley graph structures

Abstract

If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called \emph{H-integral}. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are \emph{Gaussian integers}, then the mixed graph is called \emph{Gaussian integral}. For any finite group $\Gamma$, we characterize the set $S$ for which the normal mixed Cayley graph $\text{Cay}(\Gamma, S)$ is H-integral. We further prove that a normal mixed Cayley graph is H-integral if and only if it is Gaussian integral.