  The direct product of graphs $G_1,\ldots,G_n$ is the graph with vertex set
$V(G_1)\times\cdots\times V(G_n)$ in which two vertices $(g_1,\ldots,g_n)$ and
$(g_1',\ldots,g_n')$ are adjacent if and only if $g_i$ is adjacent to $g_i'$ in
$G_i$ for all $i$. Building off of the recent work of Brakensiek, we prove an
optimal vertex isoperimetric inequality for direct products of complete
multipartite graphs. Applying this inequality, we derive a stability result for
independent sets in direct products of balanced complete multipartite graphs,
showing that every large independent set must be close to the maximal
independent set determined by setting one of the coordinates to be constant.
Armed with these isoperimetry and stability results, we prove that the upper
irredundance number of a direct product of balanced complete multipartite
graphs is equal to its independence number in all but at most $37$ cases. This
proves most of a conjecture of Burcroff that arose as a strengthening of a
conjecture of the second author and Iyer. We also propose a further
strengthening of Burcroff's conjecture.
