HS-integral and Eisenstein integral mixed circulant graphs

Abstract

A mixed graph is called second kind hermitian integral (or HS-integral ) if the eigenvalues of its Hermitianadjacency 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 a mixed circulant graph Circ(Z n , S) is HS-integral. We also show that a mixed circulant graph is Eisenstein integral if and only if it is HS-integral. Further, the eigenvalues and the HS-eigenvalues of some oriented circulant graphs are expressed in terms of generalized Möbius function.

HS-integral and Eisenstein integral mixed Cayley graphs over abelian groups

“…In [16], the authors provide an alternative proof of the characterization obtained in [23,34]. H-integral mixed circulant graphs, H-integral mixed Cayley graphs over abelian groups, H-integral normal Cayley graphs and HS-integral mixed circulant graphs have been characterized in [18], [17], [16] and [19], respectively.…”

“…Theorem 2.12. [19] Let m ≡ 0 (mod 3). Then Φ 1 m,3 (x) and Φ 2 m,3 (x) are irreducible monic polynomials in Q(ω 3 )[x] of degree ϕ(m) 2 .…”

HS-integral and Eisenstein integral normal mixed Cayley graphs

“…In particular, they also completely determined all integral Cayley graphs over the dicyclic group T 4p for a prime p. In [20], [19], [16], [18] and [17], we addressed a characterization of H-integral, HS-integral, Gaussian integral, and Eisenstein integral of mixed Cayley graphs over cyclic group and abelian group. In 2014, Godsil et al [12] characterised integral normal Cayley graphs.…”