Vietoris–Rips ultrametric condition for Laplacian approximation

Let $G$ be a finite weighted graph, let $d_{\min}$ denote the relevant minimum spectral separation from Theorem 1, and let $L_1^{(\alpha)}$ be the perturbation term in the Laplacian decomposition, measured in the Frobenius norm \left\\|L_1^{(\alpha)}\right\\|_F. For a Vietoris–Rips graph at scale $\epsilon>0$, its clusters are the connected components obtained by eliminating edges longer than $\epsilon$.

Vietoris–Rips ultrametric condition. The Vietoris–Rips clusters for some $\epsilon>0$ realise the sufficient ultrametric condition

\left\\|L_1^{(\alpha)}\right\\|_F<\frac{d_{\min}}{6}

for each cluster, as well as for the inter-cluster graph, and for some $\alpha>0$.

This condition would allow the perturbation estimate from the preceding theorem to apply locally to the cluster decomposition and to the inter-cluster graph. The source presents the assertion as plausible, but gives no resolution.