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- W3204134231 abstract "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 $Ssubseteq mathbb{Z}_n$ for which the eigenvalues $sumlimits_{kin S} w_n^{jk}$ 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 $ntimes n$ 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 $frac{1}{chi_j(1)} sumlimits_{s in S} chi_j(s)$ 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 $Ssubseteq mathbb{Z}_n$ for which the circulant digraph $Cay(mathbb{Z}_n, S)$ is Gaussian integral." @default.
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- W3204134231 date "2021-11-18" @default.
- W3204134231 modified "2023-09-27" @default.
- W3204134231 title "H-integral normal mixed Cayley graphs." @default.
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