Nullity of Hermitian-adjacency matrices of mixed graphs
Received:November 30, 2016  Revised:November 30, 2016
Key Word: nullity   mixed graph   unicyclic graph   Hermitian-adjacency matrix  
Fund ProjectL:The National Natural Science Foundation of China (General Program, Key Program, Major Research Plan)
Author NameAffiliationE-mail
凤雷 田 School of Mathematics, China University of Mining and Technology tflcumt@126.com 
Dein Wong School of Mathematics, China University of Mining and Technology  
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Abstract:
      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.
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