Let $G$ be a finite simple graph. The Lie algebra $\mathfrak{g}$ of $G$ is defined as follows: $\mathfrak{g}$ is generated by the vertices of $G$ modulo the relations $[u, v]=0$ if there is no edge between the vertices $u$ and $v$. Many properties of $\mathfrak{g}$ can be obtained from the properties of $G$ and vice versa. The Lie algebra $\mathfrak{g}$ of $G$ is naturally graded and the graded dimensions of the Lie algebra $\mathfrak{g}$ of $G$ have some deep connections with the vertex colorings of $G$. In this talk, I will explain how to get the generalized chromatic polynomials of $G$ in terms of graded dimensions of the Lie algebra of $G$. We will use this connection to give a Lie theoretic proof of of Stanleyâ€™s reciprocity theorem of chromatic polynomials.

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Last updated: 29 Mar 2023