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On generalized inverses of matrices associated with certain graph classes

Cláudia M. Araújo, Faustino A. Maciala, Pedro Patrício

TL;DR

The paper studies generalized inverses of graph‑associated matrices for two digraph classes: double star digraphs and D‑linked stars. It derives explicit Drazin index formulas and Drazin inverses for the non‑group‑invertible cases of double star matrices, and provides complete Moore‑Penrose invertibility criteria with explicit expressions wherever they exist. For D‑linked stars, it characterizes when the group inverse exists (via the invertibility of $BC$) and offers a closed form, while also establishing the Drazin index relationship in the singular BC=0 case and giving Moore‑Penrose invertibility conditions and formulas. Overall, the work links algebraic inverse structures to the underlying combinatorial graph properties, delivering unified frameworks for group, Drazin, and Moore‑Penrose inverses and facilitating practical computation through explicit block formulas.

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.

On generalized inverses of matrices associated with certain graph classes

TL;DR

The paper studies generalized inverses of graph‑associated matrices for two digraph classes: double star digraphs and D‑linked stars. It derives explicit Drazin index formulas and Drazin inverses for the non‑group‑invertible cases of double star matrices, and provides complete Moore‑Penrose invertibility criteria with explicit expressions wherever they exist. For D‑linked stars, it characterizes when the group inverse exists (via the invertibility of ) and offers a closed form, while also establishing the Drazin index relationship in the singular BC=0 case and giving Moore‑Penrose invertibility conditions and formulas. Overall, the work links algebraic inverse structures to the underlying combinatorial graph properties, delivering unified frameworks for group, Drazin, and Moore‑Penrose inverses and facilitating practical computation through explicit block formulas.

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.

Paper Structure

This paper contains 8 sections, 10 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Double star digraphs
  4. Drazin inverse
  5. Moore--Penrose inverse
  6. $D$-linked stars digraphs
  7. Drazin inverse
  8. Moore--Penrose inverse

Key Result

Lemma 2.1

Let $A$ be a square matrix. Then $i(A) = k$ if and only if $k$ is a positive integer such that $(A^k)^\#$ exists while $(A^i)^\#$ do not, for all $0< i <k$.

Theorems & Definitions (10)

  • Theorem 2.2: Cline's Theorem
  • Lemma 2.3: MacDuffee benisraelgreville, Puystjens--Robinson Puystjens
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3