The Inductive Graph Dimension from The Minimum Edge Clique Cover
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 o...
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Zusammenfassung: | 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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DOI: | 10.48550/arxiv.1903.02523 |