Convergence of spectra of digraph limits

TL;DR

This paper proves that the spectra of convergent digraphons, which are limits of directed graphs, also converge, extending spectral convergence results from undirected to directed graph limits and relating cycle densities to spectra.

Contribution

It establishes spectral convergence for digraphons and links directed cycle densities to the spectrum, extending known results from undirected graphons.

Findings

Spectra of convergent digraphons converge.

Densities of directed cycles relate to digraphon spectra.

Extends spectral convergence theory to directed graph limits.

Abstract

The relation between densities of cycles and the spectrum of a graphon, which implies that the spectra of convergent graphons converge, fundamentally relies on the self-adjointness of the linear operator associated with a graphon. In this short paper, we consider the setting of digraphons, which are limits of directed graphs, and prove that the spectra of convergent digraphons converge. Using this result, we establish the relation between densities of directed cycles and the spectrum of a digraphon.