Eccentricity integrality for two dominating runs

Establish whether eccentricity-integrality for connected threshold graphs with two dominating runs is governed by a single cubic Diophantine condition in the parameters (s_1,t_1,s_2), and determine whether that condition has infinitely many solutions.

Background

The eccentricity matrix splits off the final dominating block, reducing the first unresolved case to a 3×3 quotient matrix on the preceding blocks. This produces a cubic rather than quadratic integrality condition.

Computational evidence identifies at least one solution, (s_1,t_1,s_2)=(7,5,6), with arbitrary final-block size t_2≥1, but the claimed single-condition description and infinitude of solutions remain conjectural.

References

For $m=2$ connected threshold graphs, $\varepsilon$-integrality (equivalently, integrality of the $3\times3$ quotient matrix on blocks $S_1,T_1,S_2$ remaining after Theorem~\ref{thm:lastblock}'s reduction) is governed by a single cubic Diophantine condition on $(s_1,t_1,s_2)$, satisfied by infinitely many triples — e.g. computationally, $(s_1,t_1,s_2)=(7,5,6)$ is a verified solution, with $t_2\ge1$ arbitrary.

— The Spectral Theory of Threshold Graphs: A Survey  (2610.22838 - Manna et al., 16 Oct 2026) in Conjecture, Section 9.2