Neighborhood degree lists of graphs

TL;DR

This paper investigates the neighborhood degree list (NDL) of graphs, showing its properties, how it can distinguish graphs, and characterizing when NDLs have unique realizations, linking to threshold and difference graphs.

Contribution

It introduces the concept of NDL space connectivity, characterizes NDLs with unique realizations, and connects these to threshold and difference graphs.

Findings

NDL space is connected via switching operations.

Characterization of NDLs with unique realizations.

Link between NDLs, threshold graphs, and difference graphs.

Abstract

The neighborhood degree list (NDL) is a graph invariant that refines information given by the degree sequence and joint degree matrix of a graph and is useful in distinguishing graphs having the same degree sequence. We show that the space of realizations of an NDL is connected via a switching operation. We then determine the NDLs that have a unique realization by a labeled graph; the characterization ties these NDLs and their realizations to the threshold graphs and difference graphs.