On generalized inverses of matrices associated with certain graph classes

TL;DR

This paper studies various generalized inverses of matrices linked to specific digraph classes, providing explicit formulas and conditions for their existence, and revealing connections between algebraic inverses and graph properties.

Contribution

It offers new explicit formulas and conditions for Drazin, Moore-Penrose, and group inverses of matrices associated with double star and D-linked stars digraphs.

Findings

Explicit formulas for Drazin inverse of double star digraphs

Necessary and sufficient conditions for Moore-Penrose inverse existence

Characterization of group inverse for D-linked stars digraphs

Abstract

We investigate generalized inverses of matrices associated with two classes of digraphs: double star digraphs and D-linked stars digraphs. For double star digraphs, we determine the Drazin index and derive explicit formulas for the Drazin inverse. We also provide necessary and sufficient conditions for the existence of the Moore-Penrose inverse and give its explicit expression whenever it exists. For D-linked stars digraphs, we characterize when the group inverse exists and obtain its explicit form. In the singular case where BC = 0, we express the Drazin index of the matrix in terms of the Drazin index of the base digraph matrix. Additionally, we establish necessary and sufficient conditions for Moore--Penrose invertibility and derive explicit formulas in that case. Our results reveal a clear connection between the algebraic structure of generalized inverses and the combinatorial properties of these graph classes, providing a unified framework for group, Drazin, and Moore-Penrose invertibility.