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What are the good reading books to learn cluster algebra? I need a basic introductory books or notes in particular. I do not have any physics background and I want a book which starts with graph theory. Please help.

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At some point you will want to read Fomin and Zelevinsky's papers: I II III IV. There are also video lectures from Lauren William's course at the Institut Henri Poincaré with several resources linked on that page. There's also a draft of a textbook by Fomin, Williams and Zelevinsky on the arXiv in two parts: 1-3 4-5.

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There are some lecture notes like

1) https://math.berkeley.edu/~williams/papers/CA.pdf

2) http://www.maths.dur.ac.uk/users/philipp.b.lampe/LectureNotes/cluster.pdf

3) https://bookstore.ams.org/emszlec-19

In general, you do not need any Graph theory know to a huge extent. I have to be familiar with quivers, edges and all.

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