Tightness of upper bounds or existence of slower undirected graphs for bd updating

Determine whether the general upper bound O(N^6 log N) on the expected absorption time for neutral birth-death updating on undirected graphs is tight, or construct undirected graph families whose expected absorption time asymptotically exceeds the Θ(N^4) time scale achieved by the double star graph family.

Background

The authors prove that the double star graph has expected absorption time Θ(N^4) under bd updating, and they provide a general upper bound of O(N^6 log N) for any undirected graph. This leaves a gap between the slowest known explicit family (double stars) and the general upper bound.

Clarifying whether the double star is asymptotically optimal or whether the upper bound can be tightened—or alternatively whether even slower undirected graphs exist—would resolve this fundamental question about the extremes of diversity maintenance under bd updating.