HS-integral and Eisenstein integral normal mixed Cayley graphs

Abstract

A mixed graph is said to be second kind hermitian integral (HS-integral) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called Eisenstein integral if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set S for which the normal mixed Cayley graph Cay(Γ,S) is HS-integral for any finite group Γ. We further show that a normal mixed Cayley graph is HS-integral if and only if it is Eisenstein integral. This paper generalizes the results of Kadyan and Bhattacharjy (2022) [11].