What kind of mathematical background is needed for graph theory?
There are plenty of good books on graph theory. Start with something introductory like Graph Theory: Modeling, Applications and Algorithms, or one of the many more or less equivalent books at that level. Then move onto something more advanced, such as 


Graph Theory: An Advanced Course (Graduate Texts in Mathematics) by Adrian Bondy and U.S.R. Murty, 
Modern Graph Theory (Graduate Texts in Mathematics) by B. Bollobás or 
Graph Theory (Graduate Texts in Mathematics) by Reinhard Diestel. 


Also building on your background in discrete mathematics is invaluable. A book such as 


Concrete Mathematics: Foundation for Computer Science by Ronald L. Graham, Donald E. Knuth, Oren Patashnik


would be provide you will a solid grounding.

Most of these books would then serve as references for life.

See also this question What kind of mathematical background is needed for complexity theory?, as much of the background will overlap, in particular wrt combinatorics and stochastics (probability and statistics).