I know the chromatic symmetric function simplifies to the chromatic polynomial when 1's and 0's are subbed in for the x's. I was wondering if one could easily find the chromatic symmetric of a graph if given the chromatic polynomial of the graph?
Thanks in advance
This could only be possible if graphs with the same chromatic polynomial always had the same chromatic symmetric function. It’s well-known that graphs with the same chromatic polynomial need not be isomorphic, and indeed one of the simplest examples of non-isomorphic graphs with identical chromatic polynomials already affords an example of graphs that have different chromatic symmetric functions despite having the same chromatic polynomial.
The claw graph on four vertices, $S_3$, has proper colourings in which three vertices have the same colour, whereas the path graph on four vertices, $P_4$, has no such colourings (since that would make at least two adjacent vertices the same colour). Thus, the symmetric chromatic function of $S_3$ contains variables to the third power, and that of $P_4$ does not. But these graphs are both trees and thus both have the chromatic polynomial $(x-1)^3x$.