# The Inductive Graph Dimension from The Minimum Edge Clique Cover

## TL;DR

This paper establishes a simple recursive formula for the Knill dimension of graph joins, relates it to minimum clique covers, and derives bounds based on clique number, advancing the understanding of graph dimensions.

## Contribution

It introduces a recursive formula for the Knill dimension of graph joins and connects it to minimum clique covers, providing new insights into graph dimension calculations.

## Key findings

- Dimension of join graphs follows a simple recursive formula.
- Graphs formed by unions of complete graphs have dimension N-1.
- Bounds on Knill dimension are derived in terms of clique number.

## Abstract

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: $\mathrm{dim\,} (G_1+G_2) = 1 +\mathrm{dim\,} G_1+ \mathrm{dim\,} G_2$. 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.