2021/06/02 by Monu Kadyan, Bikash Bhattacharjya, Kadyan, Monu +1 · 1 citation
Mathematics · Chemistry · #Graph theory and applications #Synthesis and Properties of Aromatic Compounds #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2106.01261
A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. The mixed circulant graph Circ(ℤn,C) is a mixed graph on the vertex set ℤn and edge set \ (a,b): b-a∈ C \, where 0\not∈ C. If C is closed under inverse, then Circ(ℤn,C) is called a circulant graph. We express the eigenvalues of Circ(ℤn,C) in terms of primitive n-th roots of unity, and find a sufficient condition for integrality of the eigenvalues of Circ(ℤn,C). For n≡ 0 \Mod 4, we factorize the cyclotomic polynomial into two irreducible factors over ℚ(i). Using this factorization, we characterize integral mixed circulant graphs in terms of its symbol set. We also express the integer eigenvalues of an integral oriented circulant graph in terms of a Ramanujan type sum, and discuss some of their properties.