I don't know how one can tackle the following kind of question, so any hint is welcome. I formulate a precise question in order to fix ideas, but it is to be considered as an example out of a more general class.
Example: Can one embed Petersen's graph in $\mathbb R^4$ in such a way that all edges are mapped to segments (of not necessarily equal lenght) and each pair of adjacent segments forms an angle of $2\pi/3$ ?
The question can be generalized in the following way, to give a wider class of questions:
- instead of Petersen's graph, consider a given $k$-regular graph (we had $k=3$ above)
- instead of $\mathbb R^4$ consider $\mathbb R^n, n>k$, or embeddings into the round sphere $S^n$ where the edges of the graph are mapped into geodesics
- ask that the angles formed between any pair of adjacent edges are equal to the angle $VOV'$ made at which two vertices $V,V'$ of a regular euclidean $(k-1)$-simplex are seen from its barycenter $O$
I think that $1$-skeletons of spherical regular polytopes give such kind of graphs. I am interested in any partial answer which introduces essentially more ingredients than just constructions based on symmetry groups, or in obstructions which show negative answers.