Another amusing case is that of the truncated icosahedral graph (chemically referred to as Buckminsterfullerene when realized as a 60-atom carbon cluster).  Here the second eigenvalue is 3-fold degnerate, and if x, y, z denote 3 real equi-norm orthogonal eigenvectors for this eigenvalue, then the 60 triples of corresponding components locate the vertices as embedded in Euclidean space so as to manifest the icosahedral symmetry.  See: D. E. Manopolous & P. W. Fowler, J. Chem. Phys. 96 (1992) 7603-7615 .  In fact, these authors go on to similarly treat other "fullerenes" (which are polyhedra with degree-3 vertices and faces which are either pentagons or hexagons).  In general the requisite eigenvalues are not degenerate, but are those which have eigenvectors with components dividing the graph into exactly 2 connected regions of different signs for the components -- also some scaling of the components by appropriate fuctions of the different eigenvalues is used.  There has been some further work on such ideas for other suitable graphs - for the non-regular case the Laplaian matrix might plausibly be preferred over the adjacency matrix.