Diameter of 2-distance graphs

For a simple graph G, the $2$-distance graph, D₂(G), is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph G. In this paper, for graphs G with diameter 2, we show that diam(D₂(G)) can be any integer t2. For graphs G with diam(G)3, we prove that 1/2diam(G) diam(D₂(G)) and this inequality is sharp. Also, for $diam(G)=3$, we prove that diam(D₂(G))5 and this inequality is sharp.