On the Laplacian spread of digraphs

Wayne Barrett, Thomas R. Cameron, Emily Evans, H. Tracy Hall, Mark Kempton

Research output: Contribution to journalArticlepeer-review


In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validity of the Laplacian spread bound for balanced digraphs is equivalent to the Laplacian spread conjecture for simple undirected graphs, which was conjectured in 2011 and proven in 2021. Moreover, we prove an equivalent statement for weighted balanced digraphs with weights between 0 and 1. Finally, we state several open conjectures that are motivated by empirical data.

Original languageEnglish (US)
Pages (from-to)126-146
Number of pages21
JournalLinear Algebra and Its Applications
StatePublished - May 1 2023

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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