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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
© 2024 MSP (Mathematical Sciences
Publishers).