Inverse of \(α\)-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph

Let \(X\) be bipartite mixed graph and for a unit complex number \(α\), \(H_α\) be its \(α\)-hermitian adjacency matrix. If \(X\) has a unique perfect matching, then \(H_α\) has a hermitian inverse \(H_α⁻¹\). In this paper we give a full description of the entries of \(H_α⁻¹\) in terms of the paths between the vertices. Furthermore, for \(α\) equals the primitive third root of unity \(γ\) and for a unicyclic bipartite graph \(X\) with unique perfect matching, we characterize when \(H_γ⁻¹\) is \(± 1\) diagonally similar to \(γ\)-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a new construction for the \(± 1\) diagonal matrix.