Lov\'asz theta type norms and Operator Systems

TL;DR

This paper establishes a correspondence between graphs and operator systems, introduces new graph parameters via operator system norms, and explores their properties and computational methods, drawing parallels to the Lovász theta function.

Contribution

It proves that graph isomorphism corresponds to operator system isomorphism, introduces new graph parameters from operator system norms, and provides methods to compute them.

Findings

Graph isomorphism characterized by operator system isomorphism.

New graph parameters defined via quotient norms of operator systems.

Explicit semidefinite programming approach to compute these parameters.

Abstract

To each graph on $n$ vertices there is an associated subspace of the $n \times n$ matrices called the operator system of the graph. We prove that two graphs are isomorphic if and only if their corresponding operator systems are unitally completely order isomorphic. This means that the study of graphs is equivalent to the study of these special operator systems up to the natural notion of isomorphism in their category. We define new graph theory parameters via this identification. Certain quotient norms that arise from studying the operator system of a graph give rise to a new family of parameters of a graph. We then show basic properties about these parameters and write down explicitly how to compute them via a semidefinte program, and discuss their similarities to the Lov\'{a}sz theta function. Finally, we explore a particular parameter in this family and establish a sandwich theorem that holds for some graphs.