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: \(\mathrm{dim}\, (G_1+ G_2) = 1 +\mathrm{dim}\, G_1+ \mathrm{dim}...
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| description | 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. |
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| title | The Inductive Graph Dimension from The Minimum Edge Clique Cover |
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