Handshaking Theorem
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Introduction to Graphs
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Today we’re diving into graphs! What do you think a graph consists of?
Isn't it made of vertices and edges?
Exactly! The set of vertices is denoted as V, and edges are denoted as E. Remember, V must be non-empty.
What about edges? Can they be empty?
Yes, there can be graphs with no edges, but we always need vertices. Let's remember V is always non-empty!
Types of Graphs
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Now let’s distinguish between directed and undirected graphs. Who can explain?
In directed graphs, edges have a direction, right?
Correct! They are ordered pairs. In undirected graphs, edges don’t have direction.
Can you give an example of each?
Sure! A directed edge can be (u, v), and for an undirected graph, (u, v) is the same as (v, u). Remember, directed graphs show paths while undirected graphs show connections!
Handshaking Theorem
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Let’s talk about the Handshaking Theorem. What does it state about degrees of vertices?
I think it says something about the sum of the degrees being twice the number of edges.
Exactly! The sum of all vertex degrees is twice the edge count. If we denote the edges as m, then this statement is crucial.
What about vertices of odd degree? Do they follow any rules?
Great question! As per Euler's theorem, the number of vertices of odd degree must always be even. It's fundamental in graph theory!
Examples and Application
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Let's apply what we learned. Can anyone calculate the degree of a vertex?
If a vertex has three edges connecting it, its degree is three.
Correct! And how about counting edges in relation to vertex degrees?
We must add the degrees and divide by two to find the edges.
Exactly! It showcases the Handshaking Theorem in action. Keep practicing these calculations!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
This section covers the Handshaking Theorem, which highlights the fundamental relationship between the degrees of vertices and the edges in an undirected graph. It also discusses related concepts such as degree, adjacency, and implications in graph theory, including Euler's theorem regarding odd and even degrees.
Detailed
The section elaborates on the Handshaking Theorem applied to undirected graphs, asserting that the total degree (the count of edges incident to a vertex) across all vertices is always twice the total number of edges present in the graph. It explores types of graphs—directed and undirected—while defining terms like simple graphs, self-loops, incidence, and adjacency. The importance of the degree of vertices is emphasized, particularly noting that if a vertex possesses a self-loop, it counts as contributing two to its degree. The section also delves into Euler's theorem, concluding that the number of vertices with an odd degree must be even in such graphs, as derived from the theorem. The definitions of additional special types of graphs such as complete graphs, cycle graphs, and bipartite graphs are provided for context, highlighting their properties and significance within the broader study of graph theory.
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Introduction to the Handshaking Theorem
Chapter 1 of 4
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Chapter Content
Next, we state a very fundamental fact about an undirected graph this is also called as the handshaking theorem. So, if you are given an undirected graph it may be simple, it need not be simple, it is just an undirected graph. And say the graph has m number of edges. Then what the theorem basically says is that if you sum the degrees of all the vertices in the graph then it will be twice the number of edges always.
Detailed Explanation
The Handshaking Theorem states that in any undirected graph, when you add up the degrees (the number of connections) of all the vertices, the total will always be twice the number of edges. This happens because each edge connects two vertices, and thus contributes to the degree counts of both vertices involved. Therefore, every edge is counted twice in the total degree sum.
Examples & Analogies
Imagine a party where each guest (vertex) shakes hands (edges) with other guests. If you count every handshake made by each guest, you'll find that handshakes are counted for both participants. So if there are 10 handshakes in total, the degrees (the total number of handshakes counted) will sum to 20, since each one is counted for both guests.
Applying the Handshaking Theorem
Chapter 2 of 4
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Chapter Content
So, you can verify this with respect to this example graph, so you take the summation of ∑(deg(v1)) + deg(v2) + deg(v3) = 2m.
Detailed Explanation
To apply the Handshaking Theorem, you would summate the degrees of each vertex in a graph. For example, if you have three vertices (v1, v2, and v3) with degrees deg(v1), deg(v2), and deg(v3), then the total degree sum will be 2 multiplied by the number of edges (m). This allows you to find either the total number of edges or verify the number of edges if all vertex degrees are known.
Examples & Analogies
Think of a sports team where players (vertices) have assists (edges). If one player assists 2 times, another player 4 times, and another 6 times, the sum of assists is 12. According to the Handshaking Theorem, all assists have actually contributed to 6 interactions (edges), because each assist is counted for both the player making the assist and the one receiving it.
Understanding the Contribution of an Edge
Chapter 3 of 4
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Chapter Content
So, consider any arbitrary edge e = (u, v) in your undirected graph. The claim is that the contribution of this edge will be 2 to the overall summation of degrees of all the vertices.
Detailed Explanation
Each edge in an undirected graph connects two vertices and contributes to the degree count of both. For example, if you have an edge connecting vertex u and vertex v, when calculating degrees, it adds 1 to the degree of u and 1 to the degree of v. This means that each edge effectively adds 2 to the total degree count across the graph.
Examples & Analogies
Think of each handshake again. If two friends shake hands, they both count that handshake in their record of how many handshakes they've made. If we had just one handshake (edge), each friend counts it as one, making a total contribution of 2 when we tally it up.
Conclusion of the Theorem
Chapter 4 of 4
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Chapter Content
Hence it is easy to see that the summation of the degrees of all the vertices will be twice the number of edges in the graph.
Detailed Explanation
In conclusion, after accumulating the contributions of all edges, the total number of degrees will always equal twice the number of edges in the graph. This illustrates the relationship between edges and vertex degrees. This theorem can be extremely useful in various applications, such as network analysis or graph coloring, as it provides insight into the structure of the graph.
Examples & Analogies
Consider a social network where each connection represents a friendship (edge). If there are 5 friendships total, each counting for two people, that reflects how 10 people might have an average of 2 connections each. The Handshaking Theorem helps us confirm that the connections (edges) are consistently reflected in how we view the friendships (degrees).
Key Concepts
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Graph: A mathematical structure consisting of vertices and edges.
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Vertex Degree: The number of connections (edges) a vertex has.
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Handshaking Theorem: The total of all vertex degrees is equal to twice the number of edges.
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Euler's Theorem: There must be an even number of vertices with an odd degree.
Examples & Applications
Example of calculating vertex degree: If vertex A has edges connecting to B, C, and D, its degree is 3.
Example of Handshaking Theorem: In a graph with 4 vertices and 4 edges, the total degree sums to 8.
Memory Aids
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Rhymes
In a graph, edges we count, degrees to total amount; Handshaking holds the key, twice edges, it's plain to see!
Stories
Once in a town with many friends (vertices), everyone shook hands (edges). To find how many handshakes happened, count each friend’s handshakes and double the total! That's the Handshaking Theorem.
Memory Tools
For odd degrees, think ODD - Only Done at Dusk! Meaning you have an even number of odd degree vertices in play.
Acronyms
DICE
Degree In Counts of Edges
which summarizes how to calculate degrees in relation to edges.
Flash Cards
Glossary
- Graph
A collection of vertices and edges.
- Vertex
A node in a graph.
- Edge
A connection between two vertices.
- Degree of a Vertex
The number of edges incident to a vertex.
- Simple Graph
A graph without self-loops or multiple edges between vertices.
- Selfloop
An edge that connects a vertex to itself.
- Adjacent Vertices
Vertices that are connected by an edge.
- Handshaking Theorem
The sum of the degrees of vertices in a graph equals twice the number of edges.
- Euler's Theorem
In any undirected graph, the number of vertices with an odd degree is even.
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