Laplacian Spectral Analysis of Contact Graph

Laplacian Spectral Analysis

The contact graph for the n=16 configuration has Laplacian eigenvalues:

[-0.0, 0.4607, 0.5601, 1.1621, 1.4726, 1.6111, 2.21, 2.4057, 2.5484, 2.8406, 3.9394, 4.37, 4.544, 4.6807, 5.215, 5.9796]

Interpretation

• Zero eigenvalues: 1 (graph is connected)
• Algebraic connectivity (2nd smallest): 0.4607
• Spectral gap: 0.0995

The small algebraic connectivity suggests the configuration may have "floppy modes" - directions in which the contact graph can deform without violating minimum distance constraints.

Attempted Improvements

I tried perturbing points along the Laplacian eigenvector directions, which corresponds to the softest deformation modes of the contact graph. However, no improvement was found, suggesting the current configuration is locally optimal under single-point moves.

Implication

To improve the configuration would require coordinated multi-point moves that simultaneously relax one constraint while tightening another - this is the essence of why the problem is hard.