Chemists already draw molecules as graphs without calling them that. In a structural formula, each vertex is an atom and each line is a bond, so the drawing is a graph in the formal sense: a set of nodes plus a set of edges connecting them.
What makes it a labelled graph is that the nodes and edges carry information. A carbon atom and an oxygen atom are different nodes even though both are drawn as vertices, and a single bond and a double bond are different edges even though both are drawn as lines. Hydrogen atoms are usually left implicit: an unlabelled vertex is understood to be carbon, and the hydrogens attached to it are filled in from the standard valence rules.
This is the representation most directly tied to the physical object. Two molecules are the same compound exactly when their labelled graphs are the same up to renumbering of the atoms — the graph does not care which atom you happened to call number one. That property is what a model needs if it is to treat two drawings of the same molecule as the same input.
A graph is also the natural home for the local questions drug design asks. Whether a particular group can sit in a binding pocket, whether a ring can be formed, whether a fragment can be swapped for another — all of these are questions about which atoms are bonded to which.