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 $S$.
Highlights
We only consider graphs without loops and multi-edges
A graph G is denoted by G = (V (G), E(G)), where V (G) and E(G) are the vertex set and edge set of G, respectively
In a mixed graph G, we call an edge with end vertices u and v to be undirected if both (u, v) and (v, u) belong to E(G) (resp. only one of (u, v) and (v, u) belongs to E(G))
Summary
We only consider graphs without loops and multi-edges. A graph G is denoted by G = (V (G), E(G)), where V (G) and E(G) are the vertex set and edge set of G, respectively. In 1982, Bridge and Mena [6] introduced a characterization of integral Cayley graphs over abelian groups. In 2014, Cheng et al [14] proved that normal Cayley graphs (its generating set S is closed under conjugation) of symmetric groups are integral. In 2019, Cheng et al [7] obtained several simple sufficient conditions for the integrality of Cayley graphs over the dicyclic group T4n = a, b | a2n = 1, an = b2, b−1ab = a−1. We give a characterization of integral mixed Cayley graphs over abelian the electronic journal of combinatorics 28(4) (2021), #P4.46 groups in terms if its connection set. We obtain a sufficient condition on the connection set S for integrality of the mixed Cayley graph Cay(Γ, S) over an abelian group Γ.
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