On the Diameter, Girth and Coloring of the Strong Zero-Divisor Graph of Near-rings

In this paper, we study a directed simple graph ${\Gamma}_S(N)$ 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 $I,J{\in}V^*(N)$, I is adjacent to J if and only if IJ = 0. Here, we deal with the diameter, girth and coloring of the graph ${\Gamma}_S(N)$. 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 ${\Gamma}_S(N)$.