The $k$-core of a graph and its high-order spectra

TL;DR

This paper links the concept of the $k$-core in graphs with high-order spectral analysis using the $k$-adjacency tensor, establishing spectral conditions for the existence of $k$-cores and introducing a new eigenvector centrality measure.

Contribution

It introduces a spectral characterization of $k$-cores via the spectral radius of the $k$-adjacency tensor and defines a new $k$-order eigenvector centrality based on the Perron vector.

Findings

A graph has a non-empty $k$-core iff the spectral radius of its $k$-adjacency tensor is ≥ 1.

Vertices with positive Perron vector entries belong to the $k$-core.

Numerical experiments validate the spectral conditions and the centrality measure.

Abstract

The $k$-core of a graph is its largest subgraph with minimum degree at least $k$, a fundamental concept for uncovering hierarchical structures. In this paper, we establish a connection between the $k$-core and the high-order spectra of graphs, a concept originally introduced by Cvetkovi\'{c}, Doob, and Sachs. Specifically, we consider the high-order spectra defined via the $k$-adjacency tensor. Within this framework, we prove that a graph admits a non-empty $k$-core if and only if the spectral radius of the $k$-adjacency tensor is greater than or equal to $1$. Moreover, when the $k$-core exists, vertices corresponding to positive entries in the Perron vector of the $k$-adjacency tensor belong to the $k$-core. We thus define the $k$-order eigenvector centrality via the Perron vector, which provides both membership identification and a measure of relative influence within the $k$-core. Numerical experiments confirm our theoretical findings and illustrate the properties of this centrality measure in some real-world networks.