Nullity of Hermitian-Adjacency Matrices of Mixed Graphs

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 [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.