The Inductive Graph Dimension from The Minimum Edge Clique Cover

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: $\mathrm{d}\mathrm{i}\mathrm{m}(G_1+G_2)=1+\mathrm{d}\mathrm{i}\mathrm{m}{G}_1+\mathrm{d}\mathrm{i}\mathrm{m}{G}_2$. 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.