  The unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$, denoted $X_n$, is the
graph on $\{0,\dots,n-1\}$ where vertices $a$ and $b$ are adjacent if and only
if $\gcd(a-b,n) = 1$. We answer a question of Defant and Iyer by constructing a
family of infinitely many integers $n$ such that $\gamma_t(X_n) \leq g(n) - 2$,
where $\gamma_t$ denotes the total domination number and $g$ denotes the
Jacobsthal function. We determine the irredundance number, domination number,
and lower independence number of certain direct products of complete graphs and
give bounds for these parameters for any direct product of complete graphs. We
provide upper bounds on the size of irredundant sets in direct products of
balanced, complete multipartite graphs which are asymptotically correct for the
unitary Cayley graphs of integers with a bounded smallest prime factor.
