Nullity of Hermitian-adjacency matrices of mixed graphs
Nullity of Hermitian-adjacency matrices of mixed graphs

DOI：

 作者 单位 E-mail 田凤雷 中国矿业大学数学学院 tflcumt@126.com 王登银 中国矿业大学数学学院

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 [ X. Li, G. Yu, The skew-rank of oriented graphs (in Chinese), Sci. Sin. Math. 45(2015): 93-104]. 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.