Integral mixed circulant graphs

A B S T R A C T A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integers. The mixed circulant graph Circ(Zn, C) is a mixed graph on the vertex set Z(n) and edge set {(a, b) : b - a is an element of C}, where 0 (sic) C subset of Z(n). If C is closed under inverse, then Circ(Z(n), C) is called a circulant graph. We express the eigenvalues of Circ(Zn, C) in terms of primitive n-th roots of unity, and find a sufficient condition for integrality of the eigenvalues of Circ(Z(n), C). For n equivalent to 0 (mod 4), we factorize the n-th degree cyclotomic polynomial into two irreducible factors over Q(i). Using this factorization, we characterize integral mixed circulant graphs in terms of their symbol set. We also express the integer eigenvalues of an integral oriented circulant graph in terms of a Ramanujan type sum, and discuss some of their properties. (C) 2022 Elsevier B.V. All rights reserved.