The Inductive Graph Dimension from The Minimum Edge Clique Cover

Abstract

In this paper we prove that the inductively defined graph dimension has a simple additive property under the join operation. The dimension of the join of two simple graphs is one plus the sum 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 inductive dimension of an arbitrary finite simple graph from its minimum edge clique cover. A corollary of the formula is that any arbitrary finite simple graph whose maximal cliques are all of order $N$ has dimension $N-1$. We finish by finding lower and upper bounds on the inductive dimension of a simple graph in terms of its clique number.