Abstract
Let $R$ be a ring. The unitary addition Cayley graph of $R$, denoted
$\mathcal{U}(R)$, is the graph with vertex $R$, and two distinct vertices $x$
and $y$ are adjacent if and only if $x+y$ is a unit. We determine a formula for
the clique number and chromatic number of such graphs when $R$ is a finite
commutative ring. This includes the special case when $R$ is $\mathbb{Z}_n$,
the integers modulo $n$, where these parameters had been found under the
assumption that $n$ is even, or $n$ is a power of an odd prime. Additionally,
we study the achromatic number of $\mathcal{U}( \mathbb{Z}_n )$ in the case
that $n$ is the product of two primes. We prove that the achromatic number of
$\mathcal{U} ( \mathbb{Z}_{3q})$ is equal to $\frac{3q+1}{2}$ when $q > 3$ is a
prime. We also prove a lower bound that applies when $n = pq$ where $p$ and $q$
are distinct odd primes.