A higher-connectivity spectral Ore theorem for triangle-free graphs

Let $B_{n,k}$ be the graph obtained from the balanced complete bipartite graph on $n$ vertices by deleting a matching of size $k$. If $G$ is an $n$-vertex triangle-free graph with $\kappa (\text{comp}G)\ge k$, we prove that $\text{rhoA}(G)\le \text{rhoA}(B_{n,k})$ for $n\ge 4k+2$, with equality precisely when $G\cong B_{n,k}$, and we compute $\text{rhoA}(B_{n,k})$ explicitly. We also solve the bipartite problem for every $n\ge 2k+1$, determine the boundary value ${\mathrm{spex}}_{\kappa }(2k,K_3;k)=k-1$, and settle the full problem for $k=2$. In particular, $B_{n,2}$ is uniquely extremal exactly from order $6$ onward. For $k=1$, equivalently when the complement is connected, $B_{n,1}=K_{\text{ceil}n\mathrm{/}2,\text{floor}n\mathrm{/}2}-e$ is uniquely extremal for every $n\ge 3$.