Let N be a zero-symmetric near-ring with identity and let ${Gamma}(N)$ be a graph with vertices as elements of N, where two different vertices a and b are adjacent if and only if <a> + <b> = N, where <x> is the ideal of N generated by x. Let ${Gamma}_1(N)$ be the subgraph of ${Gamma}(N)$ generated by the set {n ${in}$ N : <n> = N} and ${Gamma}_2(N)$ be the subgraph of ${Gamma}(N)$ generated by the set $N{ackslash}{upsilon}({Gamma}_1(N))$, where ${upsilon}(G)$ is the set of all vertices of a graph G. In this paper, we completely characterize the diameter of the subgraph ${Gamma}_2(N)$ of ${Gamma}(N)$. In addition, it is shown that for any near-ring, ${Gamma}_2(N){ackslash}M(N)$ is a complete bipartite graph if and only if the number of maximal ideals of N is 2, where M(N) is the intersection of all maximal ideals of N and ${Gamma}_2(N){ackslash}M(N)$ is the graph obtained by removing the elements of the set M(N) from the vertices set of the graph ${Gamma}_2(N)$.