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i want to show that for an example graph family the nummer of connected subgraphs grows expnential with n.

That is easy to show for a complete graph, because a complete graph has

n(n-1)/2 = n over 2

edges. One edge is either in the subgraph or not. Therefore, every subgraph can be enumerated with a binary number of the length

2^(n over 2)

and because its a completed graph, every subgraph is connected.

But lets assume for example we want to show that the number of connected subgraphs in a 3- or 4-regular graph grows also exponential. We can enumerate the subgraphs in the same manner. But we have to exclude a lot of them, because they are not connected.

How can we do that? Is there a way to distinguish all connected subgraphs from the not connected ones?

Greetings and thanks for your thoughts

Perhalo
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1 Answers1

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This idea is easy to prove for certain families of graphs, and in particular families of graphs with a high "Edge-Connectivity" (see https://en.wikipedia.org/wiki/K-edge-connected_graph).

For an edge-connectivity greater than k, you can always choose any k vertices for removal and generate a connected graph. Hence, you get at least Summation(j = 1 .. k; E-choose-k) graphs where E is the total number of edges. Let k > (E/m) for some constant m.

Then indeed, the number of sub-graphs will grow exponentially.

Bill Province
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