One can use graphs to count Hurwitz numbers, and this technique sheds light on some beautiful combinatorial and geometric structure of double Hurwitz numbers, allowing one to re-prove and extend results of Goulden, Jackson, Vakil, Shadrin, Shapiro, and Vainshtein. I intend this talk to be a warm-up for the WAGS talk the following day, focusing on examples and aspects of the story that might be less familiar to geometers.