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In this article, we compute the regularity of Rees algebra of binomial edgeideals of closed graphs. We obtain a lower bound for the regularity of Reesalgebra of binomial edge ideals. We also study some algebraic properties of theRees algebra and special fiber ring of binomial edge ideals of closed graphsvia algebraic properties of their initial algebra and Sagbi basis theory. Weobtain an upper bound for the regularity of the special fiber ring of binomialedge ideals of closed graphs.