Harary polynomials
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Given a graph property P, F. Harary introduced in 1985 P-colorings, graph colorings where each colorclass induces a graph in P. Let χ_P(G;k) counts the number of P-colorings of G with at most k colors. It turns out that χ_P(G;k) is a polynomial in 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 P, the polynomial χ_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.
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