A best bound for $λ_2(G)$ to guarantee $κ(G) \geq 2$

Let $G$ be a connected $d$-regular graph with a given order and the second largest eigenvalue $λ_2(G)$. Mohar and O (private communication) asked a challenging problem: what is the best upper bound for $λ_2(G)$ which guarantees that $κ(G) \geq t+1$, where $1 \leq t \leq d-1$ and $κ(G)$ is the vertex-connectivity of $G$, which was also mentioned by Cioabă. As a starting point, we solve this problem in the case $t =1$, and characterize all families of extremal graphs.

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