What question we ask about the graph depends on the application, but often leads to deeper, general and abstract questions worth studying in their own right. Here is a short summary of the types of questions we have considered:
Can the graph be drawn in the plane without edges crossing? If so, how many regions does this drawing divide the plane into?
Is it possible to color the vertices of the graph so that related vertices have different colors using a small number of colors? How many colors are needed?
Is it possible to trace over every edge of a graph exactly once without lifting up your pencil? What other sorts of “paths” might a graph posses?
Can you find subgraphs with certain properties? For example, when does a (bipartite) graph contain a subgraph in which all vertices are only related to one other vertex?