λ–Core distance partitions

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