Considering network and complex systems applications such as the study of dynamics was a sufficient answer for me after learning basic graph theory. Although some of the further applications of network theory are not quite as 'clear', network theory is really where graph theory has been driven, giving way to new interesting research. Protein networks, biological networks(complex systems) and even social networks are all solid examples.
Scott Page from Michigan State has decent videos on the applications of network theor.
Although I don't have too much experience in dynamics, I can expound a little more on a popular application: social theory. Observing the properties of networks (essentially graphs) from different models such as the hub-and-spoke or Erdos/Renyi (random graph model) is usually a topic explored in network applications. An example of an interesting theorem is the 'friendship theorem' or the proof that states "your friends have more friends than you do". [Dont' worry if you don't know what I just said because it is detailed in the book by by Mark newman linked below].
Overall this strays a bit from the pure graph theory you may be used to but it gets more interesting applied to something much more tangible. Generating functions, probability, and calculus all come into play through the study of networks or changing graphs over time. If you've had complex systems you are probably familiar with not only fractals but fixed point analysis or equilibrium state analysis revolving around solving for a 'recursive' equation that outputs the value it inputs. Here it can be applied readily to this social network theory as well.
This book is a good resource if you want to know more about the real applications of graph theory. In it includes different network models that are essentially graphs (such as bipartite, simple, etc.). The 'friendship' theorem should be detailed here along with many other interesting results (although it has been a while so forgivem e if I am mistaken).
In addition googling network theory courses taught by university's serve as good resources. This is in fact the modern mathematics that is being applied in recent research. Calculus is hundred's of years old for example but network theory is relatively new considering it really got underway during the mid 1900's.
(example: ttp://mae.engr.ucdavis.edu/dsouza/ecs289) - add the h since i don't have enough rep to add more than one link.
To be honest graph theory's application is really network theory as I stated earlier. This kind of modern mathematics not only applies something as old as Konenberg's bridges but encompasses these new social network theorems detailed by Mark Newman along with modern ideas such as financial trade modeling and other extremely relvant ideas. So I would recommend looking into network applications that interest you such as medicine or whatever may be what you daydream about because chances are some crazy researcher out there is looking into modeling whatever it may be through studying the properties of networks.
A trivial example off the top of my head that I found pretty interesting was the idea of what a terrorist network is. It is something sparse without very high average degree or without high betweenness centrality or else if one person is caught, then they will be able to give information about the rest away. Thus these sparse networks serve as accurate models for large terrorsit networks since terrorists are shown to know very little about what secretive exclusive orgainization they are in. Something else that interested me that I'll leave to you to look into is disease outbreak not just through the standard differential equation model (which is essentially a complete graph view) but more of a random connection model. In addition another thing about social networks taht is interesting is the degree of seperation and the Small world experiment. I will leave that to you as well.
Finally this book by Duncan Watts is a good read for a more philosophical view on network theory if you are still not convinced.
Six Degrees: The Science of a Connected Age