H-integral and Gaussian integral normal mixed Cayley graphs

Abstract

A mixed graph is called integral if all the eigenvalues of its Hermitian adjacency matrix are integers.A mixed Cayley graph Cay(Γ, S) is called normal if S is the union of some conjugacy classes of a finite group Γ. In 2014, Godsil and Spiga characterized integral normal Cayley graphs. We give similar characterization for the integrality of a normal mixed Cayley graph Cay(Γ, S) in terms of S.Xu and Meng (2011) and Li (2013) characterized the set S ⊆ Z n for which the eigenvalues k∈S w jk n of the circulant digraph Cay(Z n , S) are Gaussian integers for all j = 1, ..., h. Here the adjacency matrix of Cay(Z n , S) is considered to be the n × n matrix [a ij ], where a ij = 1 if (i, j) is an arc of Cay(Z n , S), and 0 otherwise.Let {χ 1 , . . . , χ h } be the set of the irreducible characters of Γ. We prove that 1 χj (1) s∈S χ j (s) is aGaussian integer for all j = 1, ..., h if and only if the normal mixed Cayley graph Cay(Γ, S) is integral.As a corollary to this, we get an alternative and easy proof of the characterization, as obtained by Xu, Meng and Li, of the set S ⊆ Z n for which the circulant digraph Cay(Z n , S) is Gaussian integral.

HS-integral and Eisenstein integral mixed circulant graphs

“…In 2006, So [29] introduced characterization of integral circulant graphs. In [19], the authors provide an alternative proof of the characterization obtained in [23,35]. H-integral mixed circulant graphs, H-integral mixed Cayley graphs over abelian groups and H-integral normal Cayley graphs have been characterized in [17], [18] and [19], respectively.…”

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

“…In 2006, So [27] introduced characterization of integral circulant graphs. 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.…”

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.…”

“…In 2014, Godsil et al [12] characterised integral normal Cayley graphs. We have also offered a characterisation of H-integral normal mixed Cayley graphs in [16].…”