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