混合图的Hermitian邻接矩阵的零维数

Nullity of Hermitian-Adjacency Matrices of Mixed Graphs

中文关键词:  零维数  混合图  单圈图  Hermitian 邻接矩阵

英文关键词:nullity  mixed graph  unicyclic graph  Hermitian-adjacency matrix

中文摘要:

      图$G$称作混合图, 若$G$既包含有向边, 又包含无向边. 混合图$G$的Hermitian邻接矩阵$H(G)$的零维数,定义为$H(G)$中零特征值的个数, 记作 $\eta_H(G)$. 本文完全刻画了给定阶数和匹配数的混合单圈图的Hermitian邻接矩阵的零维数, 此结论综合了无向单圈图和定向单圈图的相关结果; 同时, 此结论修改了文献 [李学良, 于桂海. 定向图的斜秩. 中国科学: 数学, 2015, 45(1): 93--104] 中定理4.2 的结果. 此外, 本文给出了具有零维数 $n-3$ 的所有 $n$ 阶混合图, 并证明了这些混合图是由其Hermitian邻接矩阵的谱所决定的.

英文摘要:

      A mixed graph means a graph containing both oriented edges and undirected edges. The nullity of the Hermitian-adjacency matrix of a mixed graph $G$, denoted by $\eta_H(G)$, is referred to as the multiplicity of the eigenvalue zero. In this paper, for a mixed unicyclic graph $G$ with given order and matching number, we give a formula on $\eta_H(G)$, which combines the cases of undirected and oriented unicyclic graphs and also corrects an error in Theorem 4.2 of [Xueliang LI, Guihai YU. The skew-rank of oriented graphs. Sci. Sin. Math., 2015, 45: 93-104 (in Chinese)]. In addition, we characterize all the $n$-vertex mixed graphs with nullity $n-3$, which are determined by the spectrum of their Hermitian-adjacency matrices.