Two Logs For Connectivity

$O(\sqrt{m})$ amortized per update and $O(1)$ per query, from periodic rebuilds every $\sqrt{m}$ edges, since no polylogarithmic bound is known for the fully dynamic case.

$O(\log^2 n)$ amortized per update and $O(\log n / \log \log n)$ per query, from the Holm, de Lichtenberg and Thorup structure built on Euler tour trees.

$O(\log n)$ amortized per update and $O(\log n)$ per query, from a link/cut tree maintained directly over any spanning forest of the input graph.

$O(\alpha(n))$ amortized per update and $O(\alpha(n))$ per query, from a disjoint set union with rollback and union by rank over the edge set.