matrices - Max eigenvalue of symmetric matrix and its relation to diagonal values

I saw few questions about it, but still can't understand.

Let $A$ be a symmetric matrix and $\lambda_{\max}$ its largest eigenvalue. Is the following true for all $A$?

$$\lambda_{\max} \ge a_{ii} \forall i$$

That is, is the largest eigenvalue of a symmetric matrix always greater than any of its diagonal entries?

Is it somehow related to spectral radius and the following equation?

$$\rho(A)=\max|\lambda_i|.$$