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For a regular polyhedron (or polygon) centered at the origin, the coordinatesof the vertices are eigenvectors of the graph Laplacian for the skeleton ofthat polyhedron (or polygon) associated with the first (non-trivial)eigenvalue. In this paper, we generalize this relationship. For a given graph,we study the eigenvalue optimization problem of maximizing the first(non-trivial) eigenvalue of the graph Laplacian over non-negative edge weights.We show that the spectral realization of the graph using the eigenvectorscorresponding to the solution of this problem, under certain assumptions, is acentered, unit-distance graph realization that has maximal total variance. Thisresult gives a new method for generating unit-distance graph realizations andis based on convex duality. A drawback of this method is that the dimension ofthe realization is given by the multiplicity of the extremal eigenvalue, whichis typically unknown prior to solving the eigenvalue optimization problem. Ourresults are illustrated with a number of examples.

Extremal graph realizations and graph Laplacian eigenvalues