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 the chromatic symmetric function. We introduce a vertex-weighted version of XB, show that this function admits a deletion-contraction re-lation, and show that it is equivalent to a number of other vertex-weighted graph functions, namely the W-polynomial, the polychromate, and the weighted (r, q)-chromatic function. We also demonstrate that the vertex-weighted XB admits spanning-tree and spanning-forest expansions generalizing those of the Tutte poly-nomial, and show that from this we may also derive a spanning-tree formula for the chromatic symmetric function. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples. In particular, we show that there are pairs of unweighted graphs of arbitrarily high girth with equal Tutte symmetric function, and arbitrarily large vertex-weighted trees with equal Tutte symmetric function.