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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.
What is going on can be intuitively "understood", best in terms of the Laplacian, which is a discrete analog of the classical Laplacian from analysis. For this classical Laplacian, eigenvectors correspond to standing waves with energy corresponding to the eigenvalue. The long wave-length waves which have a single internal node partition the region (i.e., the graph in our analogy) with the amplitudes giving the distances from the node.
There is also W. T. Tutte's "classic" scheme for "drawing" the Schlegel diagram of a polyhedral graph: Proc. London Math. Soc. 13 (1963) 743-767. For a more recent survey on "Bridges between Geometry and Graph Theory" see T. Pisanski & M. Randic, pages 174-194 in Geopmetry at Work, ed. C. A. Gorini (MAA, Wash. DC, 2000).

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 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.  
What is going on can be intuitively "understood", best in terms of the Laplacian, which is a discrete analog of the classical Laplacian from analysis.  For this classical Laplacian, eigenvectors correspond to standing waves with energy corresponding to the eigenvalue.  The long wave-length waves which have a single internal node partition the region (i.e., the graph in our analogy) with the amplitudes giving the distances from the node.    
There is also W. T. Tutte's "classic" scheme for "drawing" the Schlegel diagram of a polyhedral graph: Proc. London Math. Soc. 13 (1963) 743-767.  For a more recent survey on "Bridges between Geometry and Graph Theory" see T. Pisanski & M. Randic, pages 174-194 in Geopmetry at Work, ed. C. A. Gorini (MAA, Wash. DC, 2000).