# Ultracontractive Properties for Directed Graph Semigroups with Applications to Coupled Oscillators

## TL;DR

This paper extends ultracontractive properties of semigroups from undirected to directed graph Laplacians, with applications to coupled oscillators, by analyzing the geometry of directed graphs and establishing sufficient conditions.

## Contribution

It introduces a framework for understanding ultracontractive properties of directed graph Laplacians and applies these results to coupled oscillator systems beyond idealized cases.

## Key findings

- Extended ultracontractive properties to directed graph Laplacians.
- Provided geometric conditions for ultracontractivity in directed graphs.
- Applied results to real-world coupled oscillator systems.

## Abstract

It is now well known that ultracontractive properties of semigroups with infinitesimal generator given by an undirected graph Laplacian operator can be obtained through an understanding of the geometry of the underlying infinite weighted graph. The aim of this work is to extend these results to semigroups with infinitesimal generator given by a directed graph Laplacian operator, through an analogous inspection of the geometry of the underlying directed graph. In particular, we introduce appropriate nomenclature to discuss the geometry of an infinite directed graph, as well as provide sufficient conditions to extend ultracontractive properties of undirected graph Laplacians to those of the directed variety. Such directed graph Laplacians can often be observed in the study of coupled oscillators, where recent work made explicit the link between synchronous patterns to systems of identically coupled oscillators and ultracontractive properties of undirected graph semigroups. Therefore, in this work we demonstrate the applicability of our results on directed graph semigroups by extending the aforementioned investigation beyond the idealized case of identically coupled oscillators.