Strong spectral property for the constructed rank-three projections

Determine whether the rank-three orthogonal projections constructed for complements of paths on at least six vertices can be obtained with the strong spectral property (SSP) by an appropriate choice of the six-vector anchor.

Background

The paper constructs rank-three orthogonal projections in the matrix pattern class of the complement of every path on at least six vertices, thereby determining the minimal multiplicity bipartition. The construction uses a six-vector anchor and an absorption argument to produce a Parseval frame, but it does not impose the strong spectral property.

Earlier results provide SSP realizations for complements of paths in certain congruence classes, whereas the paper emphasizes that SSP is a stronger requirement than merely having a rank-three positive-semidefinite two-eigenvalue realization. The authors explicitly leave unresolved whether the anchor in their construction can be selected so that the resulting projections possess an appropriate strong property, understood in context as the SSP.