My background is in group theory and algebraic geometry. I am due to give an introduction to the topos of graphs to some graph theorists. I understand the material, but what I am missing is some sort of motivating example, i.e. some statement that can be proven using category theory (I'm also fine with type theory or whatever), but that can be understood using only graph theory.
To be clear, I am not necessarily looking for a result that can only be proven using category theory. Saying "hey, you know this result from your field? Here's an alternative proof" would do just fine as a motivation.
Bonus points if the proof is low-level enough that I can actually include it in my presentation and/or if the result is well-known and/or actually useful.
I am grateful for any suggestions!