Convergence of spectra of digraph limits
Jan Grebík, Daniel Král', Xizhi Liu, Oleg Pikhurko, Julia Slipantschuk
TL;DR
This work extends the spectral-analytic framework for graphons to digraphons, showing that spectra of convergent digraphons converge in Hausdorff distance and that the directed cycle densities of a digraphon W are given by the nonzero eigenvalues via t(C_ℓ,W) = ∑_{λ≠0} m_W(λ) λ^ℓ (ℓ ≥ 3). It develops ν-convergence tools and leverages cut-norm convergence to connect limits of digraph sequences with spectral data, establishing a precise link between directed cycle densities and the limit spectrum. The results generalize known undirected-graphon facts to the asymmetric setting and provide a robust framework for analyzing directed graph limits, including extensions to digraphs with parallel opposite edges. The findings have implications for extremal combinatorics and the study of tournament limits, clarifying how spectral information of limits governs cycle densities in large directed graphs.
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.