Justification of Laplacian-Based State Ordering

Establish from first principles whether the inverted state ordering produced by the absolute-degree graph Laplacian can be physically justified, specifically whether the least colour-attractive configuration may consistently be assigned the lowest mass in the colour-Casimir tetraquark model.

Background

The paper compares two spectral constructions for the colour-weighted adjacency matrix: direct diagonalisation of the adjacency matrix and diagonalisation of an absolute-degree Laplacian. For the symmetric tetraquark graphs considered, the Laplacian reverses the ordering of the adjacency eigenvalues, assigning the lightest mass to the least attractive colour configuration rather than to the most negative, most attractive eigenvalue.

The authors regard this ordering as physically backwards because attractive colour interactions are expected to lower the mass. They abandon the Laplacian construction, but explicitly state that they have not obtained a first-principles justification for the inverted assignment. Thus, a dynamical derivation or refutation of such an assignment remains unresolved.