Nullity of Hermitian-Adjacency Matrices of Mixed Graphs
Received:November 30, 2016  Revised:September 01, 2017
Key Word: nullity   mixed graph   unicyclic graph   Hermitian-adjacency matrix
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11571360).
 Author Name Affiliation Fenglei TIAN School of Mathematics, China University of Mining and Technology, Jiangsu 221116, P. R. China Dein WONG School of Mathematics, China University of Mining and Technology, Jiangsu 221116, P. R. China
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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.