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Drazin Inverses and Walk Structure of Oriented Dutch Windmill Graphs

C. Mendes Araújo, Faustino Maciala, Pedro Patrício

Abstract

We investigate the Drazin invertibility of adjacency matrices associated with a class of oriented graphs known as oriented Dutch windmill graphs. By analyzing walks of prescribed lengths and exploiting the structure of the minimal polynomial, we obtain explicit expressions for the Drazin inverse and determine its index. The approach combines combinatorial enumeration with algebraic matrix analysis, offering a constructive characterization that generalizes known results for paths, cycles, and bipartite graphs. Beyond its intrinsic theoretical value, the framework provides insight into discrete models governed by cyclic feedback and may serve as a basis for symbolic computation of generalized inverses in structured networks.

Drazin Inverses and Walk Structure of Oriented Dutch Windmill Graphs

Abstract

We investigate the Drazin invertibility of adjacency matrices associated with a class of oriented graphs known as oriented Dutch windmill graphs. By analyzing walks of prescribed lengths and exploiting the structure of the minimal polynomial, we obtain explicit expressions for the Drazin inverse and determine its index. The approach combines combinatorial enumeration with algebraic matrix analysis, offering a constructive characterization that generalizes known results for paths, cycles, and bipartite graphs. Beyond its intrinsic theoretical value, the framework provides insight into discrete models governed by cyclic feedback and may serve as a basis for symbolic computation of generalized inverses in structured networks.
Paper Structure (5 sections, 14 theorems, 49 equations, 1 figure)

This paper contains 5 sections, 14 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Oriented Dutch windmill graphs
  3. Adjacency matrices of oriented Dutch windmill graphs
  4. Drazin index
  5. Drazin inverse

Key Result

Lemma 2.1

Let $k\in\{1,\ldots,m\}$. A walk of length $n-1$ from vertex $i$ to vertex $j$, with $i,j\in V^k$, exists in $D^m_n$ if and only if one of the following conditions holds: or or Moreover, in each of these cases, there exists exactly one such walk of length $n-1$ from $i$ to $j$ in $D^m_n$.

Figures (1)

  • Figure 1: Oriented Dutch windmill graph $D_3^4$.

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.7
  • Lemma 2.8
  • ...and 4 more