Inverse of Hermitian Adjacency Matrix of Mixed Bipartite Graphs

Mixed graph D is a graph that can be obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph is defined to be the matrix \(H=[h_{rs}]\) where \(h_{rs}=i\) if \(v_rv_s\) is an arc in D, \(h_{rs}=-i\) if \(v_sv_r\) is an arc in D, \(h_{rs}=1\) if \(v_sv_r\) is a digon in D and \(h_{rs}=0\) otherwise. In this paper we investigate when the hermitian adjacency matrix of a bipartite graph is invertible and we prove for any tree mixed graph T with invertible hermitian adjacency matrix that \(H^{-1}\) is \(\{0,\pm 1,\pm i\}\) -matrix.