Integral Mixed Cayley Graphs over Abelian Groups

Abstract

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The mixed Cayley graph $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\left\{ (a,b): b-a\in S \right\}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over an abelian group $\Gamma$ in terms of its connection set $S$.

Splitting fields of mixed Cayley graphs over abelian groups

“…As another application of Theorem 1, we give a sufficient and necessary condition for SF(Γ) = Q (or deg(Γ) = 1), that is, the mixed Cayley graph Γ = Cay(Γ, S) is Hermitian integral. [13]. Indeed, one can verify that Corollary 2 is just another form of the main result in [13].…”

“…Let Γ = Cay(G, S) with S = A ∪ B being a subset of G \ {0} such that A = −A and B ∩ (−B) = ∅. Based on a famous result of Babai [4] for calculating of the eigenvalues of Cayley color graphs, Kadyan and Bhattacharjya [13] gave an expression for the Hermitian eigenvalues of Γ in terms of the irreducible characters of G.…”

“…Lemma 1 ( [4,13]). The Hermitian eigenvalues of the mixed Cayley graph Γ = Cay(G, S) are given by γ g = λ g + µ g for g = (g 1 , .…”

“…As applications, we give the splitting fields and algebraic degrees of undirected Cayley graphs over abelian groups, which generalizes the main result of Mönius [18] for circulant graphs. Furthermore, we provide a characterization of Hermitian integral mixed Cayley graphs over abelian groups, which can be viewed as another form of the main result of Kadyan and Bhattacharjya [13].…”

HS-integral and Eisenstein integral normal mixed Cayley graphs

“…In particular, they also completely determined all integral Cayley graphs over the dicyclic group T 4p for a prime p. In [20], [19], [16], [18] and [17], we addressed a characterization of H-integral, HS-integral, Gaussian integral, and Eisenstein integral of mixed Cayley graphs over cyclic group and abelian group. In 2014, Godsil et al [12] characterised integral normal Cayley graphs.…”