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