The Inductive Graph Dimension from The Minimum Edge Clique Cover

TL;DR

This paper establishes a simple additive property of the inductive graph dimension under join operations and derives formulas for calculating the dimension based on clique covers, providing bounds related to clique number.

Contribution

It introduces a new additive property of inductive graph dimension under join and derives a formula connecting dimension with minimum edge clique cover.

Findings

Dimension of join graphs is one plus the sum of component dimensions.

Dimension can be expressed in terms of the minimum edge clique cover.

Bounds on dimension are provided in relation to clique number.

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.