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Uniformly sparse graphs and matrices

Charles R. Johnson, Wenxuan Ding and Yuqiao Li

Vol. 17 (2024), No. 2, 249–262
Abstract

The uniformly sparse (US) graphs, and the matrices associated with them, are introduced. These matrices have remarkable properties regarding diagonal equivalence and, thus, minimum rank. In the square case, this means that the determination of maximum geometric multiplicity of an eigenvalue is straightforward. Properties of US graphs, the matrices and the relationships with key parameters are discussed.

Keywords
directed graph, diagonal equivalence, minimum rank, path cover number, tree, uniformly sparse matrix, zero forcing number
Mathematical Subject Classification
Primary: 05C50
Secondary: 15A18, 15A03, 05C20
Milestones
Received: 26 February 2022
Revised: 1 July 2022
Accepted: 11 February 2023
Published: 20 May 2024

Communicated by Kenneth S. Berenhaut
Authors
Charles R. Johnson
Department of Mathematics
The College of William & Mary
Williamsburg, VA
United States
Wenxuan Ding
Department of Mathematics
The College of William & Mary
Williamsburg, VA
United States
Yuqiao Li
Department of Mathematics
The College of William & Mary
Williamsburg, VA
United States