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 staked. Thank you in advance for any help.



Connections between loops (algebraic structure) and graphs
It isn't hard to define a class of graphs whose isomorphism classes correspond to the isomorphism classes of loops. It is easiest using vertex colours: see B. D. McKay, A. Meynert and W. Myrvold, Small Latin squares, quasigroups and loops, J. Combinatorial Designs, 15 (2007) 98-119.


In [**this corrected version**](http://cs.anu.edu.au/~bdm/papers/ls_final_corr.pdf) the construction is $G\_3(L)$ on page 13. You might have to adjust it a little to achieve exactly what you need. 


