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 On the Number of Semistar Operations of some Classes of Prufer Domains Bull. Korean Math. Soc.Published online August 6, 2019 A. Mimouni KFUPM, Abstract : The purpose of this paper is compute the number of semistar operations of certain classes of finite dimensional Pr\"ufer domains. We prove that $|SStar(R)|=|Star(R)|+|Spec(R)|+|Idem(R)|$ where $Idem(R)$ is the set of all nonzero idempotent prime ideals of $R$ if and only if $R$ is a Pr\"ufer domain with $Y$-graph spectrum, that is, $R$ is a Pr\"ufer domain with exactly two maximal ideals $M$ and $N$ and $Spec(R)=\{(0)\subsetneq P_{1}\subsetneq\dots\subsetneq P_{n-1}\subsetneq M, N| P_{n-1}\subsetneq N\}$. We also characterize non-local Pr\"ufer domains $R$ such that $|SStar(R)|=7$, respectively $|SStar(R)|=14$. Keywords : Star operation, semistar operation, Prufer domain, Y -graph spectrum. MSC numbers : 13F05, 13A15. Full-Text :