# Vertex-weighted Generalizations of Chromatic Symmetric Functions

#### Abstract

Defined by Richard Stanley in the early 1990s, the chromatic symmetric function XG of a graph G enumerates for each integer partition λ of :V (G): the number of proper colorings of G that partition V (G) into stable sets of sizes equal to the parts of λ. Thus, XG is a refinement of the well-known chromatic polynomial χG , and its coefficients in different symmetric function bases provide further information on the structure of G than χG . However, XG loses some of the utility of χG because it fails to admit a natural edge deletion-contraction relation. To address this shortcoming we introduce vertex-weighted graphs (G, w) consisting of a graph G and a weight function w : V (G) → N. Then XG extends in a natural way to a new function X(G,w) on vertex-weighted graphs. We demonstrate that X(G,w) satisfies a deletion-contraction relation akin to that of the chromatic polynomial, and use this relation to both derive new properties of the chromatic symmetric function and prove previously known properties in an original way. In the case of prior results, the new proofs are typically simpler and more intuitive than the original proofs, and are more closely related to analogous proofs of properties of the chromatic polynomial. We then demonstrate how the deletion-contraction relation can be used as a new tool to research open questions involving XG . We also explore a similar extension of the bad-coloring chromatic symmetric function XBG to vertex-weighted graphs, and we consider applications of these new functions to graph isomorphism and symmetric function bases.

#### Subject Area

Mathematics

#### Recommended Citation

Crew, Logan, "Vertex-weighted Generalizations of Chromatic Symmetric Functions" (2020). *Dissertations available from ProQuest*. AAI27831338.

https://repository.upenn.edu/dissertations/AAI27831338