Question

Graduate level Graph theory

Answer 1

At graduate level, I would think, you would be looking at specialist books on algebraic graph theory, algorithmic graph theory, random graph theory, etc.

Answer 2

You could also start on the basics of Matroid Theory. Graphs are a special type of matroid ad some areas of graph theory research are done with matroids.

Answer 3

If you're interested in algebraic stuff, I think Godsil and Royal's book Algebraic Graph Theory is an awesome resource.  Lots of cool topics covered in there.  

I found Bouwer and Haemers book on spectral graph theory a little terse when I used it for a reading group, but I also really enjoyed some of the content.

Answer 4

I would advise reading Diestal if you haven't already. Also think about understanding the probabilistic method. Luke Postle has his courses on the two on YouTube.