In this paper, we study a directed simple graph Γ_s(N) for a near-ring N, where the set V^*(N) of vertices is the set of all left N-subsets of N with nonzero left annihilators and for any two distinct vertices I, J ∈ V^*(N), I is adjacent to J if and only if IJ = 0. Here, we deal with the diameter, girth and coloring of the graph Γ_s(N). Moreover, we prove a sufficient condition for occurrence of a regular element of the near-ring N in the left annihilator of some vertex in the strong zero-divisor graph Γ_s(N).

Keywords: Near-ring, N-subsets, diameter, girth, essential ideal, chromatic number, left annihilator.