In this talk, we explore the graded structure of a Leavitt path algebra L_K(E) of an arbitrary graph E over a field K. We focus on studying and constructing graded irreducible representations of L_{K}(E).

Necessary and sufficient conditions (both graphical and algebraic) are obtained under which a Leavitt path algebra has exactly one graded irreducible representation up to graded-isomorphism.

Leavitt path algebras with finite or infinite graded isomorphism classes of graded-irreducible representations are described. Examples are constructed illustrating the results. This is a joint work with Ashish Srivastava.