Basic Results of the Quantum Labeling Number

Abstract

In this paper, we introduce the reader to the graph labeling question in the setting of quantum mechanics. We formally define an edge weight of a graph G and use it to develop the general labeling number. We derive some elementary results about this number and give an example of a quantum chromatic labeling function of a simple path graph of two vertices that is not replicable in the classical setting. We define a quantum labeling number for any graph and show that it is at most as large as the classical labeling number. Finally we posit a relationship between the quantum labeling numbers as seen in different tensor products of operator spaces common to Quantum Information Theory.