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

TL;DR

This paper characterizes the inverse of the $eta$-Hermitian adjacency matrix for bipartite graphs with a unique perfect matching, providing explicit formulas and conditions for special cases involving unicyclic bipartite graphs.

Contribution

It offers a complete description of the inverse entries of the $eta$-Hermitian adjacency matrix and characterizes when this inverse is diagonally similar to a scaled adjacency matrix for unicyclic bipartite graphs.

Findings

Explicit formulas for the inverse entries in terms of paths

Characterization of when the inverse is diagonally similar to a scaled adjacency matrix

New construction method for $oxed{1}$ diagonal matrices

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.