Circulant Digraphs Integral over Number Fields

A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the sufficient and necessary condition for circulant digraphs which are integral over a number field K. And we solve the Conjecture3.3 in [XM] and find it is affirmative.Comment: 7 page

A method to determine algebraically integral Cayley digraphs on finite Abelian group

Researchers in the past have studied eigenvalues of Cayley digraphs or graphs. We are interested in characterizing Cayley digraphs on a finite Abelian group G whose eigenvalues are algebraic integers in a given number field K. And we succeed in finding a method to do so by proving Theorem 1. Also, the number of such Cayley digraphs is computed.Comment: 9 page

H-integral normal mixed Cayley graphs

A mixed graph is called integral if all the eigenvalues of its Hermitian adjacency matrix are integers. A mixed Cayley graph $Cay(\Gamma, S)$ is called normal if $S$ is the union of some conjugacy classes of a finite group $\Gamma$. In 2014, Godsil and Spiga characterized integral normal Cayley graphs. We give similar characterization for the integrality of a normal mixed Cayley graph $Cay(\Gamma,S)$ in terms of $S$. Xu and Meng (2011) and Li (2013) characterized the set $S\subseteq \mathbb{Z}_n$ for which the eigenvalues $\sum\limits_{k\in S} w_n^{jk}$ of the circulant digraph $Cay(\mathbb{Z}_n, S)$ are Gaussian integers for all $j=1,...,h$. Here the adjacency matrix of $Cay(\mathbb{Z}_n, S)$ is considered to be the $n\times n$ matrix $[a_{ij}]$, where $a_{ij}=1$ if $(i,j)$ is an arc of $Cay(\mathbb{Z}_n, S)$, and $0$ otherwise. Let $\{\chi_1,\ldots,\chi_h\}$ be the set of the irreducible characters of $\Gamma$. We prove that $\frac{1}{\chi_j(1)} \sum\limits_{s \in S} \chi_j(s)$ is a Gaussian integer for all $j=1,...,h$ if and only if the normal mixed Cayley graph $Cay(\Gamma, S)$ 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 $S\subseteq \mathbb{Z}_n$ for which the circulant digraph $Cay(\mathbb{Z}_n, S)$ is Gaussian integral