Let U be an invariant subset of finite Γ-near-ring M. There are many papers that consider the graph respect to the near-ring and the interplay between algebraic structures and graphs are studied. Indeed, it is worthwhile to relate algebraic properties of near-ring to the combinatorics properties of assigned graphs. In this paper the graph with respect to an invariant subset U of Γ-near-ring M, denoted by Γ ^{α}_{U} (M) is introduced and the basic properties of it is investigated. Also the relation between the commutativity of M and properties of this graph is presented.
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