The tautological ring of the moduli space of curves is a subring of the Chow ring that, on the one hand, contains many of the classes represented by "geometrically defined" cycles (i.e. loci of curves that satisfy certain geometric properties), on the other has a reasonably manageable structure. By this I mean that we can explicitly describe a set of additive generators, which are indexed by suitably decorated graphs. The study of the tautological ring was initiated by Mumford in the '80s and has been intensely studied by several groups of people. Just a couple years ago, Pandharipande reiterated that we are making progress in a much needed development of a "calculus on the tautological ring", i.e. a way to effectively compute and compare expressions in the tautological ring. An example of such a "calculus" consists in describing formulas for geometrically described classes (e.g. the hyperelliptic locus) via meaningful formulas in terms of the combinatorial generators of the tautological ring. In this talk I will explain in what sense "graph formulas" give a good example of what the adjective "meaningful" meant in the previous sentence, and present a few examples of graph formulas. The original work presented is in collaboration with Nicola Tarasca and Vance Blankers.