Two graphs are isomorphic if and only if their corresponding operator systems are unitally completely order isomorphic. In this talk we examine how this extra structure that comes from the operator system can be used to say something new about a graph via this correspondence. We found that 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 norms, 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 new graph theoretic condition.