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.