Theory and Applications of Graphs (Jan 2023)

HS-integral and Eisenstein integral mixed circulant graphs

  • Monu Kadyan,
  • Bikash Bhattacharjya

DOI
https://doi.org/10.20429/tag.2023.100103
Journal volume & issue
Vol. 10, no. 1
pp. 1 – 25

Abstract

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A mixed graph is called \emph{second kind hermitian integral} (\emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of the 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, we express the eigenvalues and the HS-eigenvalues of unitary oriented circulant graphs in terms of generalized M$\ddot{\text{o}}$bius function.

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