Which graph parameters are NOT concentrated on random graphs? It is well known that many important graph parameters show (strong) concentration on random graphs, at least in some range of the edge probability. Some typical examples are the chromatic number, maximum clique,  maximum independent set, maximum matching, domination number, number of copies of a fixed subgraph, diameter, maximum degree, choice number (list coloring number), Lovasz $\theta$-number, tree width, etc. 

Question: Which are the exceptions, that is, meaningful graph parameters that are not concentrated on random graphs?

Edit.
A possible definition of concentration is this:

Let $X_n$ be a graph parameter on $n$-vertex random graphs. We call it concentrated, if for every $\epsilon&gt;0$, it holds that 
$$\lim_{n\rightarrow\infty}\Pr\big((1-\epsilon)E(X_n)\leq X_n \leq (1+\epsilon)E(X_n)\big)=1.$$
The concentration is strong, if the probability approaches 1 at an exponential rate. But sometimes strong is used in a different sense, referring to the fact that the convergence remains true with a shrinking interval, yielding a possibly very narrow range. 
 For example, if $X_n$ is the minimum degree, then, for some range of the edge probability $p$,  one can prove 
$$\lim_{n\rightarrow\infty}\Pr\big(\lfloor E(X_n)\rfloor\leq X_n \leq \lceil E(X_n)\rceil\big)=1$$
which is the shortest possible interval (as the degree is integer, but the expected value may not be). 

Note: One can construct artificial exemptions from  the concentration rule. For example, let $X_n=n$, if the graph has an odd number of edges, and 0 otherwise. This is clearly not concentrated, but I would not consider it a meaningful parameter.
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.