Cores, shell indices and the degeneracy of a graph limit

TL;DR

This paper explores the extension of core, shell indices, and degeneracy from finite graphs to their dense graph limits, establishing their properties and continuity in the graphon framework.

Contribution

It introduces a normalized framework for these concepts in graph limits and proves the degeneracy's continuity with respect to the cut metric.

Findings

Degeneracy is continuous in the graphon space.

Extension of core and shell indices to graph limits.

Provides a unified framework for dense graph properties.

Abstract

The $k$-core of a graph is the maximal subgraph in which every node has degree at least~$k$, the shell index of a node is the largest $k$ such that the $k$-core contains the node, and the degeneracy of a graph is the largest shell index of any node. After a suitable normalization, these three concepts generalize to limits of dense graphs (also called graphons). In particular, the degeneracy is continuous with respect to the cut metric.