The Inductive Graph Dimension from the Minimum Edge Clique Cover

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: dim (G(1) + G(2)) = 1 + dim G(1) + 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.