Let $G$ be a graph, the line graph of $G$ denoted of $L(G)$ is defined as follows:
-The vertices of $L(G)$ are the edges of $G$
-Two vertices of $L(G)$ are adjacent iff their corresponding edges in $G$ are incident G.
Let $G$ be $d$-regular graph. For $d=2$, $G$ will be cycle graph, in this case it is easy to check that $L(G)=G$, thus $L(L(G))=G$ ($i.e.$ line graph of the line graph gives us the initial graph).
My question is for which $d\geq 3$ we have $L(L(G))=G$?
I think that $L(L(G))=G$ if and only if $d=2$, but I'm not sure how to prove it.
Any idea will be useful!