  This paper studies graphical analogs of symmetric products and unordered
configuration spaces in topology. We do so from the perspective of the discrete
homotopy theory introduced by Barcelo et al. Our first result is a
combinatorial version of a theorem of P. A. Smith, which says that the
fundamental group of any nontrivial symmetric product of $X$ is isomorphic to
$H_1(X)$. Our second result gives conditions under which the n-strand braid
group of a graph is isomorphic to its discrete analog.
