# Graphs in Discrete Math: Definition, Types & Uses

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• 0:04 Graphs in Discrete Mathematics
• 0:44 Types of Graphs
• 2:56 Use of Graphs: Example 1
• 3:46 Example 2
• 4:59 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will define graphs in discrete mathematics, and look at some different types. You'll also see how these types of graphs can be used in some real-world applications.

## Graphs in Discrete Mathematics

Mary is planning a road trip from her city to a friend's house a few cities over. There are a few different routes she has to choose from, each of them passing through different neighboring cities.

She decides to create a map. She represents the cities as points, and she puts lines between them representing the route to get from one to the other. She also includes how many miles each route is by labeling the edges with their distance.

In discrete mathematics, we call this map that Mary created a graph. A graph is a collection of points, called vertices, and lines between those points, called edges. There are many different types of graphs in discrete mathematics. Let's explore some of these.

## Types of Graphs

Though there are a lot of different types of graphs in discrete mathematics, there are some that are extremely common. Some of those are as follows:

• Null graph: Also called an empty graph, a null graph is a graph in which there are no edges between any of its vertices.
• Connected graph: A graph in which there is a path of edges between every pair of vertices in the graph. Mary's graph is a connected graph, since there is a way to get from every city on the map to every other city.
• Disconnected graph: A graph in which the path of edges does not always connect every vertex.
• Bipartite graph: A graph that can be split into two sets of vertices such that edges only go between sets, not within them.
• Weighted graph: A graph in which weights, or numerical values, are assigned to each of the edges. Mary's graph is a weighted graph, where the distances between the cities are the weights of the edges.
• Directed graph: A graph in which the edges are directed by arrows, indicating that the relationship, represented by the edge, only applies from one vertex to the other, but not the other way around. In other words, if a directed edge has an arrow from A to B, A is related to B, but B is not related to A.
• Undirected graph: A graph whose edges are not directed. Mary's graph is an undirected graph, because the routes between cities go both ways.
• Simple graph: An undirected graph in which there is at most one edge between each pair of vertices, and there are no loops, which is an edge from a vertex to itself.
• Multi-graph: A graph in which there are multiple edges between any pair of vertices or there are edges from a vertex to itself, also called a loop.
• Planar graph: A graph that can be drawn so that all of the edges of the graph do not cross each other.
• Nonplanar graph: A graph that is not a planar graph. In other words, a graph that cannot be drawn without at least one pair of its edges crossing.

Phew! That's quite a few different types of graphs and, believe it or not, there's many more. The variety shows just how big this concept is and why there is a branch of mathematics, called graph theory, that's specifically geared towards the study of these graphs and their uses. Speaking of uses of these graphs, let's take a look at a couple of examples of just that!

## Uses of Graphs: Example 1

All of the graphs we just saw are extremely useful in discrete mathematics, and in real-world applications. For example, consider Mary's road trip again. Suppose she wants to find the shortest route from her house to her friend's house. First, we list all of the different routes, then we just add up the weights of the edges in each route to get the total number of miles in each route. The one that's less than the others is the shortest route.

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