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Harary polynomials

Given a graph property $\mathcal{P}$, F. Harary introduced in 1985 $\mathcal{P}$-colorings, graph colorings where each colorclass induces a graph in $\mathcal{P}$. Let $χ_{\mathcal{P}}(G;k)$ counts the number of $\mathcal{P}$-colorings of $G$ with at most $k$ colors. It turns out that $χ_{\mathcal{P}}(G;k)$ is a polynomial in $\mathbb{Z}[k]$ for each graph $G$. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial $χ(G;k)$. We show that the characteristic and Laplacian polynomial, the matching, the independence and the domination polynomial are not Harary polynomials. We show that for various notions of sparse, non-trivial properties $\mathcal{P}$, the polynomial $χ_{\mathcal{P}}(G;k)$ is, in contrast to $χ(G;k)$, not a chromatic, and even not an edge elimination invariant. Finally we study whether Harary polynomials are definable in Monadic Second Order Logic.