  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.

Operator Algebras