Which graph parameters are NOT concentrated on random graphs?
Many parameters  of the largest connected component are not concentrated for $G(n,p)$ if $p=1/n$ and more generally if $p$ is in the critical window. Examples are the diameter and the size of the largest component, the size of the second largest component, the number of leaves the component has, etc.

See e.g.  

Aldous, David. "Brownian excursions, critical random graphs and the multiplicative coalescent." The Annals of Probability (1997): 812-854.

Nachmias, Asaf, and Yuval Peres. "Critical random graphs: diameter and mixing time." The Annals of Probability 36, no. 4 (2008): 1267-1286.

Addario-Berry, Louigi, Nicolas Broutin, and Christina Goldschmidt. "The continuum limit of critical random graphs." Probability Theory and Related Fields 152, no. 3-4 (2012): 367-406.