I would like to know whether there are known constructions which provide a bijection between loops (isomorphism classes) and (possibly directed) graphs. Any reference to a useful paper in this direction will be appreciated.
Searching combinations of words "graphs, loops, representation, bijection" the hits contain too many irrelevant links, so I get stacked. Thank you in advance for any help.
I have no clean bijection as the one you are looking for. But there is a classical link between latin squares of order $n$ and distance-regular graphs on $n^2$ vertices and diameter 2.
See
But of course, this is not the only way to consider graphs associated with loops, latin squares, orthogonal sets, ...
The literature about finite geometries, projective planes, generalized quadrangles is very instructive in this respect.