Suppose $c(G,u)$ is chromatic polynomial of connected simple graph $G$. We know that 
$|c(G,-1)|$, as Stanley proved, is the total number of directed graph on $G$, without any cycle.
Also, we know some other graphical representations of the value of $c(G,u)$.


1) Do we have any graphical representation for $|c(G,2)|$?


2) Do we have any graphical representation for the multiplicity of $2$ as a root of $c(G,u)$?


I found some graphical representation for these values, but I didn't prove them yet.


Thanks for any helpful answer and good references. 



Graphical representation of chromatic polynomial
The answer to your second question appears to be "no".


As the multiplicity of 0 is the number of connected components of a graph, and for a connected graph the multiplicity of 1 is the number of blocks, then we might hope that for a 2-connected graph, the multiplicity of "2" would be related to the number of 3-connected "parts" (in some sense).


However if this was true in any sensible fashion, then a 3-connected graph would have "2" occurring with multiplicity one and this does not happen - there are 3-connected graphs that do not satisfy this.


On the other hand, I feel that there must be SOME combinatorial interpretation, though perhaps only valid for some graphs.




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**Edit**: I looked up some of my old emails and notes and have the following conjecture, inspired by a related conjecture of Dong: 


If $G$ is a 3-connected graph such that for any independent vertex-cutset $S$, the number of components of $G-S$ is no larger than $|S|$ then $2$ is a single root of the chromatic polynomial. 


