Optimal threshold for exactness of the general graph bound

Determine the smallest threshold \(n_{0}(\delta,k,d)\) such that, for every graph with minimum degree \(\delta\), average degree \(d\), and spread parameter \(\mathrm{sp}(G,k)\), the maximized lower bound from Theorem \ref{Tm_gen} is exact whenever the graph order satisfies \(n\geq n_{0}(\delta,k,d)\), and establish whether \(n_{0}=2d+O(k+1)\).

Background

Theorem \ref{Tm_exact} proves exactness of the maximized bound for all sufficiently large graphs, but its sufficient threshold is quadratic in the parameters, specifically governed by terms of order $(2d+5(k+1)+1)^2/(4\delta)$ and $(2(d-\delta)/(k+1)+3)^2$.

The concluding problem asks for the smallest threshold that guarantees exactness uniformly over all graphs with the specified parameters. The authors report that available data suggest a substantially smaller threshold of order $2d+O(k+1)$, but they do not prove this order of growth or determine the optimal threshold.