  In this paper we prove that the recursive (Knill) dimension of the join of
two graphs has a simple formula in terms of the dimensions of the component
graphs: $\mathrm{dim\,} (G_1+G_2) = 1 +\mathrm{dim\,} G_1+ \mathrm{dim\,} G_2$.
We use this formula to derive an expression for the Knill dimension of a graph
from its minimum clique cover. A corollary of the formula is that a graph made
of the arbitrary union of complete graphs $K_N$ of the same order $N$ will have
dimension $N-1$. We finish by finding lower and upper bounds on the Knill
dimension of a graph in terms of its clique number.
