We explore the concept of a graph homomorphism through the lens of C * -algebras and operator systems. We start by studying the various notions of a quantum graph homomorphism and examine how they are related to each other. We then define and study a C *algebra that encodes all the information about these homomorphisms and establish a connection between computational complexity and the representation of these algebras. We use this C * -algebra to define a new quantum chromatic number and establish some basic properties of this number. We then suggest a way of studying these quantum graph homomorphisms using certain completely positive maps and describe their structure. Finally, we use these completely positive maps to define the notion of a "quantum" core of a graph.

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Abstract. To each graph on n vertices there is an associated subspace of the n × 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 Lovász theta function. Finally, we explore a particular parameter in this family and establish a sandwich theorem that holds for some graphs.

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