| Summary: | In the last few decades, studies of graph theory based on algebraic structures have been pursued. Many studies have
inspected graphs via groups, rings, near-rings, and seminear-rings, and vice versa. In our study, first we introduce different type
of υ-primary ideals (υ = 0, 1, 2, 3, c), which generalize υ-prime ideals (υ = 0, 1, 2, 3, c) in near-rings. Then, we provide some
important characterizations of these newly initiated ideals. In addition, we explore some relationships among these ideals.
Throughout, we furnish our results with appropriate examples. Finally, as an application, we provide characterizations of
different graphs associated with these υ-primary ideals (υ = 0, 1, 2, 3, c) in near-rings.
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