Let $G=(V,E)$ be a finite undirected graph on $n$ vertices and $m$ edges. The line digraph of $G$, denoted $L(G)$, is an associated directed graph that captures edge incidences of $G$. More precisely, interpret each undirected edge of $G$ as two directed edges so that $G$ is now a directed graph. Then the vertices of $L(G)$ are the $nm$ directed edges, and a vertex (corresponding to directed edge $\vec{e}$) is connected to a vertex (corresponding to directed edge $\vec{e}'$) if $\vec{e}$ feeds into $\vec{e}'$ in $G$.
My question is regarding the structure of $G$ given the structure of $L(G)$.
Suppose $G$ is a Cayley graph for the group $\Gamma$ and symmetric generating set $S$, then can $L(G)$ also be interpreted as a Cayley digraph of some group and (non-symmetric) generating set?
EDIT: In light of Prof.Godsil's answer and the references therein, the answer to this is NO. The automorphism groups of $G$ and $L(G)$ are the same, and so since $L(G)$ is larger than $G$ (except when $G$ is a path or cycle), $L(G)$ is not a Cayley graph.
But the converse question is:
Conversely, suppose the line digraph $L(G)$ of a regular graph $G$ is a Cayley digraph of some group and generating set. Then is $G$ a Cayley graph of some group and symmetric generating set? What can we say about its group and generating set?
Would removing all digons from the line digraph (thus making it a non-backtracking digraph for the underlying graph) change the situation into a more tractable one?
Firstly I am not very clear on when a directed graph arises as a line digraph of a simple graph. As in, I know of the general results of Harary in this regard, but is there a simpler picture when the given digraph is a Cayley digraph?
