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arXiv · 2112.08085

HS-integral and Eisenstein integral mixed circulant graphs

Monu Kadyan·Bikash Bhattacharjya
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Abstract

A mixed graph is called \emph{second kind hermitian integral}(or \emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called \emph{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 $\text{Circ}(\mathbb{Z}_n, S)$ 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 M$\ddot{\text{o}}$bius function.

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Monu Kadyan, Bikash Bhattacharjya. 2021-12-15. HS-integral and Eisenstein integral mixed circulant graphs. https://arxiv.org/abs/2112.08085

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