Regular Graphs: “Clique Insertion”

March 12, 2008

After talking about the basics of regular graphs, let me now introduce something more interesting. When we make changes to a regular graph topology, like addition or removal of edges or nodes, the resulting topology might not remain regular. An interesting question then is the following: are there ways to modify a given regular graph topology, while maintaining its regularity?

Here I introduce a simple operation called “clique insertion” that “extends” a graph (or makes a graph bigger by adding more nodes) while preserving the regularity of the graph. The basic idea of this is due to sri. Later on I developed it a bit more and now it looks like it is a useful result.

Given an r-regular graph of n nodes, the graph can be extended to an r-regular graph of n + (r – 1) nodes by a process of “node insertions” (or “clique insertion”).

Explanation: Let us start with the minimally connected non-trivial regular graph: a circle with n nodes. We can see that by “inserting” a node between any two nodes, we can extend the graph into a bigger graph while preserving the regularity. In other words, cut an arbitrary edge into two, and make their loose ends incident on a newly added node. We can go on inserting any number of nodes in this fashion. We just get bigger circles, all of which have a regularity of 2.

 

It is pretty simple to see in a circle. Now, let’s go to a slightly more complicated topology. Consider a 3-regular graph. We can get a bigger 3-regular graph by inserting 2 nodes (as explained above), and adding another edge between the 2 nodes.

 

It is equivalent to inserting a clique of exactly r – 1 (where r is the regularity) nodes in the existing graph (refer the figure below). For an arbitrary n and arbitrary r, we can construct an n + (r – 1) node graph by inserting exactly r – 1 nodes, and adding edges among them to form a r – 1 node clique.

 

                                                               

In the figure, we have a 4-regular graph (incidentally, it is the complete graph K5). We now do a “clique insertion” operation on the graph to get a new 4-regular graph of 8 nodes. The nodes marked in brown form the clique.

 

The above result can be extended thus: given an r-regular graph of n nodes, we can construct a graph of n + m * (r – 1), where m is a positive integer. So, we can easily construct arbitrarily large regular graphs of a given regularity r by inserting any number of cliques of size exactly r – 1.

 

 

 

Usefulness: This has an important connotation in distributed systems, where nodes joining and failures are frequent. Several distributed hash tables (DHTs) require a symmetric degree distribution. They also need to accommodate for node joining and leaving.

  • For one, this result can be developed to be a measure the amount of disruption in symmetry that can be expected when changes in topology occur.
  • Also, it can be used to design topology changes so as to preserve symmetry. Perhaps, by using dummy nodes.

Question: Is there a similar way to make a graph smaller while preserving the regularity?

That was one example of preserving a structural property of a graph. It is an interesting to find out if we can have a whole set of such operators. We can look at several other properties as well: diameter, average path length, centrality distributions, connectivity etc.. I shall define this problem more formally in another post later on.

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Regular Graphs: Bounds

March 12, 2008

In this post, I discuss a few results about connected regular graphs that are fairly obvious. However, to the best of my knowledge they have not been discussed explicitly in literature.

  • If a connected graph of n (n > 2) nodes is regular, then it is at least 2-regular. In other words, the lower bound on the regularity r of a connected graph of more than 2 nodes, is 2.

Proof: Let us start with a minimally connected graph of n > 2 nodes. It is a straight line graph with n−1 edges, in which every node, except the two end nodes of the straight line, is connected to a “predecessor” and a “successor”. To make this regular, we add one more edge between the two end nodes of the straight line. The graph now is a circle, with every node connected to 2 other nodes, thus making it the minimal connected regular graph.

  • The upper bound on the regularity r of a n node (n > 0) graph is n − 1.

Proof: In a graph of n nodes, every node can connect to a maximum of n − 1 nodes, forming a complete graph. Hence the above result follows.

  • As a corollary of the above result, we have: the lower bound on the number of nodes n required to construct a r regular graph is r + 1.