Graph-theoretic lower bounds for homological regularity

Determine whether the regularity of \(H_i(B\Gamma)\) as an \(F[E]\)-module admits a lower bound expressed in terms of graph-theoretic properties of a general graph \(\Gamma\), including vertex and edge counts or connectivity invariants, for every fixed homological degree \(i\ge 2\).

Background

The paper identifies the onset of polynomiality with regularity questions for the homology modules over the edge-polynomial ring F[E]F[E]. For homological degrees at least 2, only limited computational and special-case information is available. The authors therefore ask whether graph structure can provide general lower bounds for these regularities.

References

For a general graph \Gamma=(V,E) and i\ge 2, is there an expression for a lower bound of the regularity of H_i(B\Gamma) in terms of graph-theoretic properties of \Gamma such as the numbers of vertices and edges, some connectivity invariants, and so on?

— Hilbert polynomials of configuration spaces over graphs of circumference at most 1  (2505.24416 - An et al., 30 May 2025) in Section 1, subsection “Future directions,” item 3