Convergence of graphs with intermediate density

TL;DR

This paper introduces a new graph convergence concept bridging bounded degree and dense graphs, analyzing spectral properties and graph parameters, and proposing novel limit objects called graphonings.

Contribution

It develops a unified framework for graph convergence, introduces graphonings as generalizations of graphons and graphings, and studies their properties and applications.

Findings

Spectra and graph parameters behave well under the new convergence.

Explicit asymptotics for coloring numbers in large girth sequences.

Construction of Euclidean graphonings for hypercubes and projective planes.

Abstract

We propose a notion of graph convergence that interpolates between the Benjamini--Schramm convergence of bounded degree graphs and the dense graph convergence developed by L\'aszl\'o Lov\'asz and his coauthors. We prove that spectra of graphs, and also some important graph parameters such as numbers of colorings or matchings, behave well in convergent graph sequences. Special attention is given to graph sequences of large essential girth, for which asymptotics of coloring numbers are explicitly calculated. We also treat numbers of matchings in approximately regular graphs. We introduce tentative limit objects that we call graphonings because they are common generalizations of graphons and graphings. Special forms of these, called Hausdorff and Euclidean graphonings, involve geometric measure theory. We construct Euclidean graphonings that provide limits of hypercubes and of finite projective planes, and, more generally, of a wide class of regular sequences of large essential girth. For any convergent sequence of large essential girth, we construct weaker limit objects: an involution invariant probability measure on the sub-Markov space of consistent measure sequences (this is unique), or an acyclic reversible sub-Markov kernel on a probability space (non-unique). We also pose some open problems.