Generalize modular and Galois properties to other graph families

Determine how the modular and Galois-theoretic properties established for independence polynomials of iterated strong products of cycle graphs extend to more general lattices, including toroidal grids and triangular or hexagonal tilings.

Background

The paper analyzes modular collapses, orbit structures, and Galois-theoretic features of independence polynomials for iterated strong products of cycles. It identifies these properties as consequences of the interaction between graph symmetries, modular arithmetic, and the algebraic structure of polynomial roots.

The authors explicitly ask whether analogous behavior persists for other lattice-like graph families, specifically naming toroidal grids and triangular or hexagonal tilings. Resolving this would test the scope of the paper’s methods beyond strong products of cycles.

References

Generalizing to Other Graph Families: How do the modular and Galois-theoretic properties we observed extend to more general lattices (e.g. toroidal grids, triangular or hexagonal tilings)?

— Algebraic and Combinatorial Stability of Independence Polynomials in Iterated Strong Products of Cycles  (2503.07910 - Hildebrant, 10 Mar 2025) in Section 6, “Future Directions,” item 1

How do the modular and Galois-theoretic properties we observed extend to more general lattices (e.g.\ toroidal grids, triangular or hexagonal tilings)?

— Algebraic and Combinatorial Stability of Independence Polynomials in Iterated Strong Products of Cycles  (2503.07910 - Hildebrant, 10 Mar 2025) in Section 6, “Future Directions” (Section \ref{sec:future-research})