On the Laplacian spread of digraphs

TL;DR

This paper extends the concept of Laplacian spread to directed graphs using the restricted numerical range, providing bounds, exact values for certain families, and connecting to longstanding conjectures in graph theory.

Contribution

It introduces a new approach to Laplacian spread in digraphs, establishes sharp bounds, and links the problem to the Laplacian spread conjecture for undirected graphs, including weighted cases.

Findings

Laplacian spread values for specific digraph families

Sharp upper bounds for polygonal and balanced digraphs

Equivalence between bounds for balanced digraphs and undirected graphs

Abstract

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.