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arXiv · 2205.07010

Inverse of $α$-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph

Mohammad Abudayah·Omar Alomari·Omar AbuGhneim
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Abstract

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 $\pm 1$ diagonally similar to $γ$-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a new construction for the $\pm 1$ diagonal matrix.

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Mohammad Abudayah, Omar Alomari, Omar AbuGhneim. 2022-05-14. Inverse of $α$-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph. https://arxiv.org/abs/2205.07010

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