I am a rising senior in a small liberal arts college, and I was wondering if anyone could suggest me good graduate schools for graph theory. My only exposure to graph theory has been the intro graph theory course at my school, so I clearly do not know much about the different varieties of graph theory, but I was wondering if anyone could shed some light on the graduate schools which provide research focus on any of the varieties of graph theory.
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The University of Waterloo has an entire Faculty of Mathematics, consisting of several departments: Pure Math, Applied Math, Combinatorics & Optimization, Statistics, Operations Research, and (at least when I went there) Computer Science. Anyway, the Department of C&O has several people that work on graph theory, as you can see here: http://math.uwaterloo.ca/combinatoricsandoptimization/research/areas#graphtheory 


First of all, let me point out that I am not a graph theorist. I have been told by a graph theorist that UCSD has very strong faculty specifically in graph theory. It also seems to me that Rutgers has some very strong combinatorialists and graph theorists. Ultimately you should decide on a place based on who you would like and are likely to work with. EDIT: A small remark regarding Matt comment. I've seen two types of people in graduate school. Those that arrive not knowing what they want to do and those that arrive because they knew already what they want to do and with whom. People in the second category typically have already done research in the field and start working on some questions right away. People in the first category take more time, take some course, and find out what they like. Of course there are exceptions to everything. 


I was a part of the Budapest Semesters in Math way back in 2007, and many of the others who studied in that program went on to phds in graph theory (I was the black sheep, going for algebraic topology instead). Several went to CEU in Budapest, and loved it. You can work with any professor in the city, and as you can imagine there are some very powerful graph theorists there (Laszlo Lovasz springs to mind). One or two also ended up at USC (that's South Carolina, not southern California) and worked with Szemeredi. If I was going for graph theory I think those two would be high on my list. Another interesting idea is University of Chicago to work with Laszlo Babai. He's technically computer science but has supervised mathematicians before. By the way, I second Waterloo and Rutgers 


Another option would be Northeast university, which has a ph.d. program on network science (basically applied graph theory). Barabasi and Vespigniani are faculty there, and if you have had little exposure, you may want to consider a more practical approach 


Here's my (incomplete) list of recommendations based on my current knowledge of graph theory research. General: Rutgers, UCSD, Tel Aviv, Waterloo, McGill, Princeton, Yale, Eötvös Loránd University, Rényi Institute, Emory University, Queen Mary University of London, Warwick University, Simon Frasier University, Charles University in Prague, University of Hamburg. Algebraic Graph Theory: Waterloo, University of Western Australia, Tilburg University, CWI Amsterdam, QMUL, Imperial College London, Tohoku University, Delaware, Clemson, Michigan Technological University, Wyoming. 

