While reading on graph theory and its applications, I came across a problem which stated if $\bar{G}$ is the complement of a regular graph then its eigenvalues are $-\lambda_i-1$ and $n-1-k$ where $G$ is $k$-regular. I know the proof for that theorem however I wanted to know does the inequality below generally hold?
Let $G$ be a graph and $\lambda_2$ its second largest eigenvalue, denote the smallest eigenvalue of $\bar{G}$ by $\lambda_n(\bar{G})$ is it true that:
$$\lambda_2+1 \ge -\lambda_n(\bar G)$$
I have tried using Rayleigh quotient but couldn't go further than that, hints are appreciated.