Hermitian-adjacency matrices and Hermitian energies of mixed graphs
Hermitian-adjacency matrices and Hermitian energies of mixed graphs
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DOI：
10.1016/j.laa.2014.10.028
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发表时间：
2015-02
影响因子：
1.1
通讯作者：
Jianxi Liu;Xueliang Li
中科院分区：
文献类型：
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作者：
Jianxi Liu;Xueliang Li
本文引入了混合图的复邻接矩阵，它是一个厄米特矩阵，称为厄米特邻接矩阵。它同时包含了无向图的邻接矩阵和有向图的斜邻接矩阵。研究了它的一些性质。进一步研究了其特征多项式的性质。研究了混合图之间的共谱问题，包括混合图与它们的基图、定向图与它们的基图。给出了混合图（特别是有向图）与其基础图共享同一谱的等价条件。因此，我们重新确认了崔和侯在文献[8]中提出的一个猜想。我们还证明了混合图的Hermitian矩阵的谱在改变其任意割边（如果有的话）时是不变的，相应地，我们引入了混合图的能量，称之为Hermitian能量.它同时包含了无向图的能量和有向图的斜能量。给出了它的一些界。特别地，刻画了具有最优厄米特能量上界的混合图.构造了一类具有最大厄米特能量的混合图.此外，混合树的厄米能量被证明是等于它的基础树的能量。最后给出了混合图的厄米特能量的积分公式。
A complex adjacency matrix of a mixed graph is introduced in the present paper, which is a Hermitian matrix and called the Hermitian-adjacency matrix. It incorporates both adjacency matrix of an undirected graph and skew-adjacency matrix of an oriented graph. Some of its properties are studied. Furthermore, properties of its characteristic polynomial are studied. Cospectral problems among mixed graphs, including mixed graphs and their underlying graphs, oriented graphs and their underlying graphs, are studied. We give equivalent conditions for a mixed graph (especially oriented graph) that share the same spectrum with its underlying graph. As a consequence, we reconfirm a conjecture which was proposed by Cui and Hou in Ref. [8]. We also show that the spectrum of the Hermitian matrix of a mixed graph is invariant when changing the value of any its cut edge (if any).Correspondingly, an energy of a mixed graph is introduced and called the Hermitian energy. It incorporates both the energy of an undirected graph and the skew energy of an oriented graph. Some of its bounds are given. Especially, the mixed graphs with optimal upper bound of Hermitian energy are characterized. An infinite family of mixed graphs attaining the maximum Hermitian energy is constructed. Moreover, the Hermitian energy of a mixed tree is showed to be equal to the energy of its underlying tree. Finally, the integral formula for Hermitian energy of a mixed graph is given.