Posted by admin | Posted in Uncategorized | Posted on 01-04-2011
Tags: design, graph regular, graph regular expressions, graph regularity lemma, graph regularization, graph regularization for color image processing, programming, regex, tools, visualization
Graph Regular
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Graph Theory: "K-Regular" Graphs?
Hey guys! I have this other question on my test review sheet that I don't know how to answer. This is exactly how it's written:
2. A graph is called "k-regular" if every vertex has degree of exactly k. How many different non-isomorphic 7-regular graphs on 10 vertices are there? Make sure to provide a justification for your answer.
I was thinking that I have to use the complement graph for this question, but I'm not really sure. I know I don't HAVE to use it, but I thought I might say what I was thinking.
Please help! Thanks So Much!!!
I think you're on to something with the "complement graph" idea.
For 10 nodes [easier to type than vertices], the complete graph
has every node connected to 9 others, so it's 9-regular.
7 is only 2 shy of that so let's look at how we might remove edges.
If we remove edges and make another graph of them, then
we see (clearly ?) those are 1-1 with the 7-regular graphs
and they're much easier to count.
A 2-regular graph looks like this (for however many nodes):
o --- o --- ...--- o --- o
| . . . . . . . . . .. . . .|
o --- o --- ...--- o --- o
In other words, a ring.
If we remove 2 edges from each node of the big graph,
and make a new one, we form some rings of various sizes.
Each one has to have at least 3 nodes, otherwise we have multiple connections.
So the 10 can be broken up into:
10
7 + 3
6 + 4
5 + 5
3 + 3 + 4
I think that's all of them. So 5 different graphs.
If it had asked about 5-regular or 6-regular or 4-regular
it would have been much more difficult.
Checking:
Complete graph has 10 * 9 /2 = 45 edges
7-regular graph has 10 * 7 / 2 = 35 edges
We must remove 10
All the above graphs have 10 edges,
since in a ring (which is 2-regular), # edges = # nodes.
