This (i.e. the non-increase of $\lambda_2$ w.r.t. edge removal) follows from the fact that $$\lambda_1=\max_{x\geq 0,\|x\|=1} x^\top A x.$$

The latter is a consequence of the characterisation of $\lambda_1$ as the maximum Rayleigh quotient and the positivity of the corr. eigenvector.

This (i.e. the non-increase of $\lambda_1$ w.r.t. edge removal) follows from the fact that $$\lambda_1=\max_{x\geq 0,\|x\|=1} x^\top A x.$$

The latter is a consequence of the characterisation of $\lambda_1$ as the maximum Rayleigh quotient and the positivity of the corr. eigenvector.