In this work I discuss the $sigma-$function of a graph. I will discuss the $sigma-$function and the $\vartheta-$function of Lov\'asz, cliques, coloring, and much more. I will show the results of experiments (pictures!), give some basic structure theory, and mention conjectures and open problems. If time permits I will talk about applications to graph drawing and clique finding. This talk should be accessible to all. This is joint work with Fan Chung.