Let $G$ be a (finite, simple, connected) graph. Define the distance-$k$ graph $G_k$ to be the graph with the same vertex set and $x\sim y$ iff $d(x,y)=k$. A graph is integral if all of the eigenvalues of its adjacency matrix are integers.
I have been looking at regular integral graphs. I noticed that when I constructed the distance-2 graphs of these graphs, they were also regular and integral. The regularity part makes sense to me, but I don't have any intuition or understanding of why the integrality might be preserved under this operation.
I haven't been able to find too much about distance-$k$ graphs online other than at http://mathworld.wolfram.com/Distancek-Graph.html. Do you know of any results relating the spectrum of $G_2$ (or of $G_k$) to the spectrum of $G$? General facts about $G_k$ graphs would also be of interest to me.