$\lambda$-Core Distance Partitions

TL;DR

This paper introduces a novel vertex partition based on $ore$ eigenvector entries and neighborhoods, linking graph structure to spectral properties and proposing an entropic measure for graph information content.

Contribution

It defines $ore$ distance partitions, explores their structural implications, and introduces an entropic measure related to eigenvalues of the universal adjacency matrix.

Findings

Partition relates graph structure to spectral properties

Entropic measure quantifies graph information content

Results connect eigenvalues to graph partitioning

Abstract

The $\lambda$-core vertices of a graph correspond to the non-zero entries of some eigenvector of $\lambda$ for a universal adjacency matrix $\mathbf{U}$ of the graph. We define a partition of the vertex set $V$ based on the $\lambda$-core vertex set and its neighbourhoods at a distance $r$, and give a number of results relating the structure of the graph to this partition. For such partitions, we also define an entropic measure for the information content of a graph, related to every distinct eigenvalue $\lambda$ of $\mathbf{U}$, and discuss its properties and potential applications.