Diameter Constraints in 2-distance Graphs

For any finite, simple graph $G=(V,E)$, its $2$-distance graph $G_2$ is a graph having the same vertex set $V$ where two vertices are adjacent if and only if their distance is $2$ in $G$. Connectivity and diameter properties of these graphs have been well studied. For example, it has been shown that if $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(G)=k\ge 3$ then $\lceil \frac{1}{2}k\rceil \le \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(G_2)$, and that this bound is sharp. In this paper, we prove that $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(G_2)=\mathrm{\infty }$ (that is, $G_2$ is disconnected) or otherwise $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(G_2)\le k+2$. In addition, we show that this inequality is sharp for any even $k$, a result that we verify for some higher orders through judicious use of a $\text{textsc}sat$ solver.