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I am interested in graph theory. My background is mainly algebraic. I have been researching algebraic geometry for five years so I assume that the transition to the graph theory realm shouldn't be so hard. I would like to ask you for a royal path to study advanced graph theory on my own and get a grasp of what are the problems that people are interested in nowadays. I know that this question is broad and general, so any help or advice would be welcome.

Thank you for your time.

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    $\begingroup$ Why the downvote? There have been similar, well-received, questions at MO. $\endgroup$
    – efs
    Oct 6, 2021 at 14:17
  • $\begingroup$ Since you have a background in algebra, you can perhaps read the nice little book "Algebraic graph-theory" by Biggs. A lot of algebraic background is for example useful in expander-graphs. You can perhaps also consider problems in Cayley-graphs or shift to hyperbolic groups (and their Cayley graphs). $\endgroup$ Oct 6, 2021 at 14:51
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    $\begingroup$ sidorenko's conjecture, ramsey's theorem bounds, triangle removal lemma bounds $\endgroup$ Oct 6, 2021 at 17:24
  • $\begingroup$ This is an excellent resource: Lovász, László. Graphs and Geometry. Vol. 65. American Mathematical Soc., 2019. And there is a PDF version linked from his Budapest webpage. $\endgroup$ Oct 6, 2021 at 18:16

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