Line Line Line Graph

Consider an undirected graph $G$ with $N$ vertices and $M$ edges. A *line graph* $L(G)$ of this graph is a graph with $M$ vertices; each of these vertices corresponds to an edge in $G$ and two vertices in $L(G)$ are connected by an edge if the corresponding edges in $G$ have a common vertex. Let $L^k(G)$ denote the result of performing the operation "replace $G$ by $L(G)$" $k$ times, i.e. $L^1(