Sharp subcritical largest-component bound

Determine the tight upper bound, in terms of the maximum degree cDelta(G) and spectral radius clambda(G), for the typical size of the largest component of G_p when p cleq (1-cepsilon)/clambda(G), for graphs with prescribed cDelta(G) and clambda(G).

Background

Theorem 1.3 gives a general subcritical upper bound whose scale depends on |G|, cDelta(G), and clambda(G), while Theorem 1.2 improves this in the regime clambda(G) comparable to csqrt{cDelta(G)} by removing a logarithmic factor. The paper also notes sharper behavior when clambda(G) is constant or when a connected graph satisfies clambda(G)=cDelta(G), the latter implying regularity and an O(log |G|) bound.

These examples show that the available estimate is not uniformly optimal across all relationships between maximum degree and spectral radius. The open question asks for the correct component-size scale throughout the full parameter range, including the nearly regular regime.

References

It would be interesting to know the correct upper bound for all regimes of $\Delta(G)$ and $\lambda(G)$. \begin{question}\label{question:tight subcritical} For a graph with given $\Delta(G)$ and $\lambda(G)$, what is the tight upper bound for the typical size of the largest component of $G_p$, when $p \leq (1-\varepsilon)/\lambda(G)$? \end{question}

— The critical probability for percolation on finite graphs  (2608.19145 - Christoph et al., 19 Aug 2026) in Question 2, Section 6, Concluding remarks (question:tight subcritical)