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A short class with Sarah and her students. The other ways are under More.
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Today, we will discuss what a graph is. A graph consists of a set of vertices and a set of edges. Can anyone tell me what a vertex is?
Isn't a vertex just a single point or node in the graph?
Exactly! And edges connect these vertices. Now, in an undirected graph, edges are simply lines connecting these points without a direction. Can someone provide an example of how to represent a simple undirected graph?
A simple undirected graph might just have points A and B, connected by a line.
Correct! We represent them as (A, B), and here, (B, A) would be the same edge. So, remember: in undirected graphs, edge direction doesn’t matter!
Now let's discuss adjacency. If we have two vertices connected by an edge, we say they are adjacent. For example, if we have vertices A and C connected, how are they related?
They are neighbors!
Exactly! And what about the degree of a vertex? How would you define that?
The degree of a vertex is the number of edges connected to it.
Well said! Remember, if there’s a self-loop, it counts as two towards the vertex's degree. If vertex A has a self-loop and one edge to B, its degree would be 3.
Now, let’s introduce the handshaking theorem. This theorem states that the sum of the degrees of all vertices is twice the number of edges in the graph. Can anyone explain what this means?
So if we sum all the degrees of each vertex, it should equal two times the total number of edges?
Exactly! This holds true for all undirected graphs regardless of their complexity. This fact helps us understand many properties within graph theory.
Why does it equal twice the edges?
Good question! Each edge connects two vertices, contributing to the degree count of both vertices.
Next is Euler's theorem, which tells us that in any undirected graph, the number of vertices with an odd degree is always even. Can anyone think of why that would be?
Because each edge adds to two vertex degrees, right? So the odd degrees will have to balance out to make an even count!
Precisely! Thus, it’s impossible to have an odd number of odd degree vertices.
What if all the vertices are even?
That’s also fine! The count of odd degree vertices can be zero as well. Great thinking!
Let’s conclude with special types of undirected graphs. Can anyone name one type?
A complete graph!
Correct! In a complete graph, there is one edge between each pair of distinct vertices. How about bipartite graphs?
They are divided into two sets with edges only between the sets.
Exactly! All vertices in one set connect to all vertices in the other, but not within their own set. Excellent discussion today!
A structure made of vertices and edges connecting them.
A node in a graph.
A link connecting two vertices.
Vertices connected directly by an edge.
Count of edges connected to a vertex.
The sum of vertex degrees is double the edges.
The number of odd degree vertices is even.
Each pair of vertices is connected by an edge.
Divided into two sets where edges only connect vertices between sets.
Example of a graph: Imagine a social network where each person is a vertex, and each friendship is an edge.
For a complete graph with three vertices (A, B, C), edges are (A, B), (A, C), (B, C), totaling three edges.
A bipartite graph could be a group of students and their grades where edges only connect students to grades.
In a graph you see, points called vertices, edges connect, like a happy family!
Imagine a city where people (vertices) are connected by roads (edges), and everyone can meet each other if all roads exist—this is like a complete graph.
Remember 'AED' for Degrees: A for Adjacency, E for Edges counted, D for Degree of a vertex.
Think 'BEC' for Bipartite Edge Connection, where edges only go from one set of vertices to another.
Term
What is a graph?
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What is the degree of a vertex?
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Define a complete graph.
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What is a bipartite graph?
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A collection of two sets: a set of vertices and a set of edges.
A node in a graph.
A connection between two vertices.
Two vertices connected by an edge.
The number of edges incident to a vertex.
An edge that connects a vertex to itself.
The theorem stating the sum of degrees of vertices in a graph is twice the number of edges.
The theorem stating that the number of vertices with an odd degree is always even.
A graph where every pair of distinct vertices is connected by an edge.
A graph that can be divided into two disjoint vertex sets such that every edge connects a vertex from one set to a vertex in the other set.
This section introduces fundamental terminologies related to undirected graphs, including definitions of graphs, simple graphs, degrees of vertices, and special types of undirected graphs.
In this section, we explore the basic terminologies associated with undirected graphs. We cover concepts such as vertices, edges, adjacency, and the degree of a vertex. Moreover, we delve into key theorems like the handshaking theorem and Euler's theorem, and introduce several special types of undirected graphs such as complete graphs and bipartite graphs.
In this section on undirected graphs, we define key terms essential for understanding graph theory. A graph is fundamentally characterized by its set of vertices (nodes) and edges (connections), with undirected graphs lacking edge directionality. Important definitions include 'adjacent vertices,' where two vertices are considered neighbors if an edge connects them, and 'degree of a vertex,' which counts edges incident to a vertex, including double counting for self-loops. We introduce the handshaking theorem, which states that the sum of degrees across all vertices in a graph equals twice the number of edges — a foundational result in graph theory. Furthermore, Euler's theorem regarding the number of vertices of odd degree is discussed. Special types of undirected graphs such as complete graphs, cycle graphs, wheel graphs, n-cubes, and bipartite graphs illustrate varied structures within graph theory. Each of these types has unique properties that contribute to the broader understanding of graphs.