A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of chromatic function. We introduce vertex-weighted version XB and show that this admits deletion-contraction relation. also demonstrate spanning-tree spanning-forest expansions generalizing those polynomial by connecting to other graph functions. Second, give several methods for constructing nonisomorphic graphs with equal functions, use them provide specific examples.