Distinguishability and linear independence for $H$-chromatic symmetric functions

We study the $H$-chromatic symmetric functions $X_G^H$ (introduced in (arXiv:2011.06063) as a generalization of the chromatic symmetric function (CSF) $X_G$), which track homomorphisms from the graph $G$ to the graph $H$. We focus first on the case of self-chromatic symmetric functions (self-CSFs) $X_G^G$, making some progress toward a conjecture from (arXiv:2011.06063) that the self-CSF, like the normal CSF, is always different for different trees. In particular, we show that the self-CSF distinguishes trees from non-trees with just one exception, we check using Sage that it distinguishes all trees on up to 12 vertices, and we show that it determines the number of legs of a spider and the degree sequence of a caterpillar given its spine length. We also show that the self-CSF detects the number of connected components of a forest, again with just one exception. Then we prove some results about the power sum expansions for $H$-CSFs when $H$ is a complete bipartite graph, in particular proving that the conjecture from (arXiv:2011.06063) about $p$-monotonicity of $\omega(X_G^H)$ for $H$ a star holds as long as $H$ is sufficiently large compared to $G$. We also show that the self-CSFs of complete multipartite graphs form a basis for the ring $\Lambda$ of symmetric functions, and we give some construction of bases for the vector space $\Lambda^n$ of degree $n$ symmetric functions using $H$-CSFs $X_G^H$ where $H$ is a fixed graph that is not a complete graph, answering a question from (arXiv:2011.06063) about whether such bases exist. However, we show that there generally do not exist such bases with $G$ fixed, even with loops, answering another question from (arXiv:2011.06063). We also define the $H$-chromatic polynomial as an analogue of the chromatic polynomial, and ask when it is the same for different graphs.