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Let $G$ be a connected $d$-regular graph with a given order and the secondlargest eigenvalue $\lambda_2(G)$. Mohar and O (private communication) asked achallenging problem: what is the best upper bound for $\lambda_2(G)$ whichguarantees 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 astarting point, we solve this problem in the case $t =1$, and characterize allfamilies of extremal graphs.

A best bound for $\lambda_2(G)$ to guarantee $\kappa(G) \geq 2$