Drazin Inverses and Walk Structure of Oriented Dutch Windmill Graphs

TL;DR

This paper studies the Drazin inverse of adjacency matrices for a special class of oriented graphs called Dutch windmill graphs, combining combinatorial and algebraic methods to derive explicit formulas and insights.

Contribution

It provides a new explicit characterization of the Drazin inverse for oriented Dutch windmill graphs, extending known results for simpler graph classes.

Findings

Explicit formulas for the Drazin inverse are derived.

The index of the Drazin inverse is determined.

The approach generalizes to other structured networks.

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.