Speaking of industrial applications, graph theory is quite popular among computer people. For example Facebook's EdgeRank determines who sees what in the news feed. And neo4j is a popular (or at least well-advertised) graph database, with gremlin a graph-traversal language. LinkedIn has a team devoted to "social network analysis" and graphs are hot as well in "complexity theory"theory" including models of the brain. For example http://video.neo4j.org/RHqy/the-pathology-of-graph-databases-by-marko-a-rodriguez/ gave me a flavour of how the web-programming crowd sees graphs.
I don't know if any of the above interests you but at least outside of pure mathematics I believe quivers-as-directed-graphs have practical applications.
