On the Dual Drazin Inverse of Adjacency Matrices of Dual-number-Weighted Digraphs

TL;DR

This paper investigates the dual Drazin inverse of adjacency matrices in dual-number-weighted digraphs, providing explicit formulas, extending previous results, and solving open problems in the field.

Contribution

It derives explicit formulas for the dual Drazin inverse of certain dual-number-weighted digraphs and extends existing results to new classes of digraphs.

Findings

Explicit formulas for dual Drazin inverse of dual complex anti-triangular block matrices.

Weakened assumptions for DN-DS digraphs compared to previous work.

Extended group inverse results to dual Drazin inverse for DN-DW digraphs.

Abstract

The motivation of this paper is to investigate the dual Drazin inverse of adjacency matrices arising from several classes of connected dual-number-weighted digraphs over the dual complex algebra. Explicit formulas for the dual Drazin inverse of dual complex anti-triangular block matrices are derived under suitable assumptions. These results are applied to DN-DS digraphs, DN-DLS digraphs, and DN-DW digraphs. In particular, the assumptions in \cite{AMPMJM2026} are weakened for DN-DS digraphs, an open problem in \cite{AMPMJM2026} for the case $BC=0$ is generalized and solved for DN-DLS digraphs. And the group inverse result in \cite{MNSEJAA2022} for bipartite block form adjacency matrices is extended to the dual Drazin inverse for DN-DW digraphs. We further derive explicit formulas for the dual group inverse and dual Drazin inverse of another adjacency matrix for DN-DW digraphs.