# Graphs formed of vertices of distance $2$

Let $$G=(V,E)$$ be a finite, simple, undirected graph. Let $$D_2(G)$$ be the graph with vertex set $$V$$, and two vertices form an edge if and only if they have distance $$2$$ in the original graph $$G$$.

Note that, if $$C_{2k+1}$$ is the circle on $$2k+1$$ points for an integer $$k\geq 2$$, we have $$C_{2k+1}\cong D_2(C_{2k+1})$$.

Question. Is there a connected, non-regular graph $$G=(V,E)$$ with $$|V|>2$$ such that $$G\cong D_2(G)$$?

Yes, the hexagon ABCDEF with the additional edge of AC has this property. The first graph shows $$G$$ and the others show two views of $$D_2(G)$$.