Extension of the analysis to the 6×6 SNIEP

Investigate whether residual regions analogous to the order-five impossibility region can be identified for the $6\times6$ symmetric nonnegative inverse eigenvalue problem and whether the diagonal-shift, complementary-commutation, and support-reduction methods developed for order five remain useful in that setting.

Background

The paper establishes a new impossibility region for the 5×55\times5 SNIEP by shifting a hypothetical realizing matrix to a critical high-trace boundary, constructing a complementary nonnegative commuting matrix, and reducing its support to a weighted five-cycle. These techniques exploit structural features specific to order five.

The 6×66\times6 SNIEP is identified as the natural next unresolved dimension. The paper asks whether analogous residual regions and related structural proof techniques exist in that higher-order problem, without resolving either question.

References

Several questions remain open. First, the complementary region $\setminus$ remains unclear, and it would be interesting to obtain a finer description of this region, either through additional impossibility conditions or through new realization constructions. Second, the $6\times6$ SNIEP is a natural next case: one may ask whether analogous residual regions can be identified and whether the shift, commutation, and support-reduction ideas developed here remain useful in higher dimensions.

— A New Impossibility Region for the $5\times5$ Symmetric Nonnegative Inverse Eigenvalue Problem  (2608.19435 - Jin et al., 19 Aug 2026) in Section Discussion