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arXiv · 1903.02523

The Inductive Graph Dimension from The Minimum Edge Clique Cover

Kassahun Betre·Evatt Salinger
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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.

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Kassahun Betre, Evatt Salinger. 2019-03-06. The Inductive Graph Dimension from The Minimum Edge Clique Cover. https://arxiv.org/abs/1903.02523

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