Degree (of a Vertex)

Some mathematical terms have multiple meanings. One example is the word "degree," which can refer to concepts in geometry, graph theory, or equations. Let's take a look at the geometrical definition of this term today and learn what a degree really is:

What is a degree?

In graph theory, we use the word degree to describe the number of edges connected to a vertex. As you might recall from your geometry knowledge, a vertex is a shape's "point." A triangle has three vertices, for example.

In the world of graph theory, a vertex serves a similar but slightly different purpose. While a vertex of a triangle is simply a point, a vertex in graph theory is usually represented by a circle, often with a label. This label may identify the vertex but does not necessarily show the number of edges connected to it.

So what exactly is an edge? An edge is a line that connects vertices together in graph theory. The best way to understand the degree of a vertex in graph theory is by viewing a visual representation:

As we can see, each vertex has a certain number of edges connected to it. If we want to find a vertex's degree, all we need to do is count the number of connected edges. For example, one vertex has five edges connected to it, so it has a degree of 5. All of the other vertices have just one connected edge, which means that they have a degree of 1. We often call these vertices "end vertices" or "leaves" because they represent the end of a "pathway."

Degree sequences

A degree sequence is a non-increasing list of the degrees of vertices in a graph. In other words, it is a list that shows the degrees of each vertex in descending order. Degree sequences are a useful way to characterize graphs and make comparisons between them.

To better understand degree sequences, let's consider a simple example. Imagine we have a graph with the following degrees for its vertices: 4, 3, 3, 2, 1, 1. We can represent this as a degree sequence: (4, 3, 3, 2, 1, 1). This tells us that the graph has one vertex with degree 4, two vertices with degree 3, one vertex with degree 2, and two vertices with degree 1.

Degree sequences can be used to determine certain properties of a graph. For instance, the sum of the degrees in a degree sequence should be an even number. This is because each edge connects two vertices, so every time we count an edge for one vertex, we also count it for another vertex. This property is known as the Handshaking Lemma.

Why does the degree of a vertex matter?

If we understand the degree of a vertex, we can solve real-world problems using graph theory. A notable example of one of these problems was "Euler''s Walk." The Swedish mathematician Euler was tasked with plotting a path through Königsberg while crossing each of its seven bridges once and only once. The task was impossible due to the way the bridges were laid out in the city. He was able to determine the impossibility of this task by creating a visual representation of the bridges and the bodies of water they traversed. This was one of the first forms of graph theory, and Euler was able to prove the impossibility of the task partly by counting the degrees of each bridge or vertex.

Topics related to the Degree (of a Vertex)

Graph Theory

Endvertex (Graph Theory)

Complex Numbers

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