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For a field K, Leavitt path algebras over a directed graph E are the natural generalization of the algebras investigated by Leavitt in [1962, The module type of a ring]. Leavitt path algebras have been introduced by Gene Abrams and Gonzalo Aranda Pino in [2005, The Leavitt path algebra of a graph] and independently by Ara, Moreno, Pardo in [2007, Non-stable K- theory of Graph algebras] as algebraic analogues of a graph C*-algebras. Results for graph C*- algebras have guided the investigation of Leavitt path algebras. A version of Cuntz-Krieger uniqueness theorem is proved by Mark Tomforde in [2007, Uniqueness theorem and ideal structure for Levitt path algebras]. In this talk, I will briefly give the required definitions in the theory of Leavitt path algebras and describe graded versus non-graded prime ideals of Leavitt path algebras. Main scope of this talk is to describe when a two-sided ideal of is an intersection of primitive/prime ideals in . [This is joint work with Müge Kanuni and Kulumani M. Rangaswamy]