1. Graph Theory
What is a graph?

2. Brief History
It all began in 1736 when Leonhard Euler gave a proof that not all seven bridges over the Pregolya River could all be walked over once and end up where you started. This became known as the “Konigsberg Bridge Problem”
First used to solve puzzles and analyze games but then was used for things like the “Four Color Map Conjecture”

3. What is a graph?
A finite set of vertices (V) and edges (E)

5. Types of graphs
Directed graph: a graph that has edges with specific directions
Multigraph: a graph with parallel edges
Pseudograph: a graph with loops
Simple graph: a graph with no loops or parallel edges

6. Definitions of Parts of Graphs
Order: the number of vertices a graph has
Size: the number of edges a graph has
Adjacent vertices: vertices that share an edge
Degree of a vertex: number of edges that vertex has

7. Degree Theorem
In a graph G, the sum of the degrees of the vertices is equal to twice the number of edges
Or the number of vertices with odd degree is even

8. Example 2
Find the degree of each of the following graphs:

9. More Definitions…..
Connected graphs: every pair of vertices can be reached from another one
Cut vertex: removal of the vertex causes the graph to become disconnected
Bridge: removal of an edge causes the graph to become disconnected

10. Types of graphs, con’t
Complete graphs: a graph where all vertices share an edge
Empty graph: a graph with no edges
Complement: a graph that contains all the edges that are not present in the original graph
Regular graphs: graphs where every vertex has the same degree

11. Example 3 Draw a complete graph with 4 vertices Draw the complement of
Draw a graph with a cut vertex and a bridge