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

Abstract: Let $G$ be a connected $d$-regular graph with a given order and the second largest eigenvalue $\lambda_2(G)$. Mohar and O (private communication) asked a challenging problem: what is the best upper bound for $\lambda_2(G)$ which guarantees that $\kappa(G) \geq t+1$, where $1 \leq t \leq d-1$ and $\kappa(G)$ is the vertex-connectivity of $G$, which was also mentioned by Cioab\u{a}. As a starting point, we solve this problem in the case $t =1$, and characterize all families of extremal graphs.