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Today, we're discussing regular graphs. A simple graph is classified as regular if all vertices have the same degree. Can anyone tell me what we mean when we use the term 'degree' in graph theory?
I think the degree of a vertex is the number of edges connected to it.
Absolutely correct! And if all vertices share the same degree r, we call it an r-regular graph. For example, the complete graph K_n with n vertices is n-1 regular because each vertex connects to every other vertex. Can someone give me another example of a regular graph?
Maybe the cycle graph?
Yes, perfect! A cycle graph has a degree of 2 for each vertex. Remember this: K_n means 'complete', and cycle graphs are shaped like loops.
Now that we understand regular graphs, let’s look at some examples. We’ve mentioned complete graphs and cycle graphs. What about the wheel graph? Anyone know its characteristics?
The wheel graph has a central node connected to outer nodes, right? I think it’s not regular because the center has more connections.
Exactly! The center node has a degree significantly higher than the others, making it irregular. To remember: if a graph has a 'hub' that impacts the rest, it's likely non-regular. Would anyone like to recapitulate what makes a graph regular?
All vertices must have the same degree!
Next, let’s construct a graph. If we're tasked with creating a simple regular graph where each vertex has a degree of 2k + 1, how might we start?
Could we use bipartite graphs?
Good thinking! Using complete bipartite graphs and combining them can work. Let’s say we have two partitions of 2k nodes each. By connecting each node in one partition to the other, we can ensure each has a degree of 2k.
But how do we ensure they each have a degree of 2k + 1?
Great question! We add additional edges from a special vertex, linking it to every node in at least one of the partitions. This creates the needed degree. Remember: custom edges can adjust degrees!
Defined as graphs where all vertices have the same degree.
Important for identifying the nature (regular vs. irregular) of a graph.
Include complete graphs and cycle graphs; some like wheel graphs are not regular.
Involves strategic addition of edges to meet specific degree requirements.
In a graph that’s regular, all degrees the same, that makes it special, it's part of the game.
Imagine a town where every person knows exactly 3 others. They live in a regular, friendly neighborhood wherein all buildings have connections planned equally.
KCC - K for Complete, C for Cycle, C for Cut edges to remember types of graphs.
RADI - Regular At Degree Identical.
Term
What is a regular graph?
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Term
Give an example of a non-regular graph.
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Term
What is a complete graph?
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A graph in which all vertices have the same degree.
The number of edges incident to a vertex.
A graph where every pair of distinct vertices is connected by a unique edge.
A graph that consists of a single cycle, where each vertex has degree 2.
A graph resembling a wheel, with one central node connected to all outer nodes.
A graph whose vertices can be divided into two disjoint sets such that every edge connects a vertex from one set to the other.
This section introduces concepts of regular graphs and explores examples of different types of regular graphs.
In this section, the definition of regular graphs is explained, along with examples such as complete graphs, cycle graphs, and non-regular graphs like wheel graphs. The focus extends to constructing a regular graph with specific degrees and cut edges.
A regular graph is defined as a simple graph where every vertex has the same degree. If this common degree is denoted as r, the graph is referred to as an r-regular graph. Key examples include: