Graph-algebras-Faithful representations and mediating objects in mathematics.

I consider the role of diagrams in contemporary mathematics. More specifically the role of certain diagrams-so-called directed graphs-will be investigated. I propose that these graphs act as mediating objects. This means that they link certain objects, that is, a C*-algebra and its associated K-groups, and that this link yields an epistemic gain. I explain that the link is possible because a graph represents as a metaphor in two distinct ways. In addition, the diagrammatic presentation of a directed graph becomes an object that can be manipulated. As such, it becomes what I will denote a "faithful representation." The notion of a faithful representation tries to capture the fruitfulness of the combination of metaphorical representation with the possibility of controlled manipulation.