λ–Core distance partitions

Abstract

The λ–core vertices of a graph correspond to the non–zero entries of some eigenvector of λ for a universal adjacency matrix U of the graph. We define a partition of the vertex set V based on the λ–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 λ of U, and discuss its properties and potential applications.