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Chromatic polynomial of a graph $G$ is an important tool in Graph theory which has been studied extensively from graph theory perspective as well as through other area of Mathematics also. Hence it is natural to wonder about the various ways it can be expressed in terms of notions in the respective areas. I am interested in Knowing some of such expressions. I have listed few of such expression below :

  1. https://arxiv.org/abs/1303.1148 contains an expression for the chromatic polynomial interms of the root multiplicities of Kac Moody Algebras.

  2. http://cours.xavierviennot.org/Talca_2013_14_files/Talca13%3A14_Ch6.pdf contains an expression for chromatic polynomials from combinatorics view point by Prof.Xavier Viennot

3.At y=0, the Tutte polynomial specialises to the chromatic polynomial https://en.wikipedia.org/wiki/Tutte_polynomial#Chromatic_polynomial which can be thought of as one among lot of Graph theoritic expression. I am aware of few more but I could not find the proper links.

I am sure there are plenty of such expression coming from various branches of Mathematics, please feel free to add them. Thanks for your interest.

Thanks for your valuable timing.

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    $\begingroup$ You might want to look into the Chromatic symmetric polynomials (which extends the chromatic polynomials) and how they sometimes relate to Hessenberg varieties. See Shareshian and Wachs. $\endgroup$
    – Per Alexandersson
    Commented Aug 31, 2016 at 16:38
  • $\begingroup$ @PerAlexandersson oh.nice. I only know about Tutte polynomial as a generalisation of Chromatic polynomial. This looks interesting. thanks for the reference too. $\endgroup$
    – GA316
    Commented Aug 31, 2016 at 16:43
  • $\begingroup$ The original introduction of the chromatic symmetric polynomial is due to R. Stanley. There are some cool open problems in this area. $\endgroup$
    – Per Alexandersson
    Commented Aug 31, 2016 at 18:26

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