2 Removed paragraph not directly relevant to this question.

Yes: All adjacency vectors can be sensibly interpreted.

I'll repeat an answer I gave to another another question ...

A matrix whose rows form an orthogonal basis of an eigenspace of a graph's adjacency matrix has columns that serve as coordinate vectors of an harmonious geometric realization of the graph.

("harmonious" == automorphisms of the graph induce rigid isometries the realization)

When the graph has a high degree of symmetry, these realizations --which I call "spectral"[*]-- have a great visual appeal; in general, though, these realizations are jumbles of points in one-dimensional space. In most cases, multiple vertices (and edges) are collapsed to single points, so that the realizations aren't faithful.

If a graph happens to admit a faithful spectral realization, you might be able to tease out a cell structure (which may not be unique), though I've not investigated this. I'll note that, even for polyhedra, there's no guarantee that the "faces" of a spectral realization are bounded by planar cycles of edges.

[*] More precisely, the realizations as described here are orthogonal projections of realizations I call "spectral". (See my still-drafty note, "Spectral Realizations of Graphs", the bulk of which is dedicated to a gallery of many, many spectral realizations of the uniform polyhedra. Also see my Mathematica Demonstration "Spectral Realizations of Polyhedral Skeleta".)

1

Yes: All adjacency vectors can be sensibly interpreted.

I'll repeat an answer I gave to another another question ...

A matrix whose rows form an orthogonal basis of an eigenspace of a graph's adjacency matrix has columns that serve as coordinate vectors of an harmonious geometric realization of the graph.

("harmonious" == automorphisms of the graph induce rigid isometries the realization)

When the graph has a high degree of symmetry, these realizations --which I call "spectral"[*]-- have a great visual appeal; in general, though, these realizations are jumbles of points in one-dimensional space. In most cases, multiple vertices (and edges) are collapsed to single points, so that the realizations aren't faithful.

If a graph happens to admit a faithful spectral realization, you might be able to tease out a cell structure (which may not be unique), though I've not investigated this. I'll note that, even for polyhedra, there's no guarantee that the "faces" of a spectral realization are bounded by planar cycles of edges.

[*] More precisely, the realizations as described here are orthogonal projections of realizations I call "spectral". (See my still-drafty note, "Spectral Realizations of Graphs", the bulk of which is dedicated to a gallery of many, many spectral realizations of the uniform polyhedra. Also see my Mathematica Demonstration "Spectral Realizations of Polyhedral Skeleta".)