Distinctive power and comparability of Harary polynomial

Abstract: Let $\mathcal{P}$ be a graph property. A $\mathcal{P}$-coloring with at most $k$ colors is a coloring of the vertices of a simple graph $G$ such that each color class induces a graph in $\mathcal{P}$. Harary polynomials are generalizations of the chromatic polynomial for simple graphs based on conditional colorings. We denote by $χ{\mathcal{P}}(G; k)$ the number of $\mathcal{P}$-colorings of $G$ with at most $k$ colors. $χ{\mathcal{P}}(G; k)$ is a polynomial in $\Z[k]$. A first paper studying Harary polynomials systematically was published in 2021 by O.Herscovici, J.A. Makowsky and V. Rakita. It studies under which conditions on $\mathcal{P}$ is $χ{\mathcal{P}}(G; k)$ definable in Monadic Second Order Logic and under which conditions is $χ{\mathcal{P}}(G; k)$ a chromatic invariant. Let $\mathcal{P}, \mathcal{Q}$ be two graph properties. Two graphs $G, H$ are $\mathcal{P}$-mates if $χ{\mathcal{P}}(G; k) = χ{\mathcal{P}}(H; k)$. $χ{\mathcal{Q}}$ is at least as distinctive as $χ{\mathcal{P}}$, $χ{\mathcal{P}} \leq χ{\mathcal{Q}}$, if for all graphs $G, H$ we have that $χ{\mathcal{Q}}(G; k) = χ{\mathcal{Q}}(H; k)$ implies $χ{\mathcal{P}}(G; k) = χ{\mathcal{P}}(H; k)$. In this paper we study under which conditions on $\mathcal{P}$ are there any (many) $\mathcal{P}$-mates and under which conditions on $\mathcal{P}, \mathcal{Q}$ is $χ{\mathcal{Q}}$ is at least as distinctive as $χ{\mathcal{P}}$.