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A short class with Sarah and her students. The other ways are under More.
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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!
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!
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!
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!
A mathematical structure consisting of vertices and edges.
The number of connections (edges) a vertex has.
The total of all vertex degrees is equal to twice the number of edges.
There must be an even number of vertices with an odd degree.
In a graph, edges we count, degrees to total amount; Handshaking holds the key, twice edges, it's plain to see!
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.
For odd degrees, think ODD - Only Done at Dusk! Meaning you have an even number of odd degree vertices in play.
DICE: Degree In Counts of Edges, which summarizes how to calculate degrees in relation to edges.
Term
Handshaking Theorem
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Even Degree Vertices
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Odd Degree Vertices
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Degree Calculation
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A collection of vertices and edges.
A node in a graph.
A connection between two vertices.
The number of edges incident to a vertex.
A graph without self-loops or multiple edges between vertices.
An edge that connects a vertex to itself.
Vertices that are connected by an edge.
The sum of the degrees of vertices in a graph equals twice the number of edges.
In any undirected graph, the number of vertices with an odd degree is even.
The Handshaking Theorem states that in any undirected graph, the sum of the degrees of all vertices equals twice the number of edges.
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.
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.