Inverse of $\alpha$-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph

Abstract

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