The $k$-core of a graph and its high-order spectra
Abstract
The -core of a graph is its largest subgraph with minimum degree at least , a fundamental concept for uncovering hierarchical structures. In this paper, we establish a connection between the -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 -adjacency tensor. Within this framework, we prove that a graph admits a non-empty -core if and only if the spectral radius of the -adjacency tensor is greater than or equal to . Moreover, when the -core exists, vertices corresponding to positive entries in the Perron vector of the -adjacency tensor belong to the -core. We thus define the -order eigenvector centrality via the Perron vector, which provides both membership identification and a measure of relative influence within the -core. Numerical experiments confirm our theoretical findings and illustrate the properties of this centrality measure in some real-world networks.
Keywords
Cite
@article{arxiv.2512.05351,
title = {The $k$-core of a graph and its high-order spectra},
author = {Chunmeng Liu and Qing Xu and Changjiang Bu},
journal= {arXiv preprint arXiv:2512.05351},
year = {2025}
}