1. CHAPTER 15
Graph Theory

2. Graphs, Paths, and Circuits
15.1
Graphs, Paths, and Circuits

3. Objectives
Understand relationships in a graph.
Model relationships using graphs.
Understand and use the vocabulary of graph theory.

4. Graphs
A graph consists of a finite set of points, called vertices, (singular is vertex), and line segments or curves, called edges, that start and end at vertices. An edge that starts and ends at the same vertex is called a loop.

5. Example 1: Understanding Relationships in Graphs
Explain why the figures below show equivalent graphs.
Solution: In both figures, the vertices are A, B, C, and D. Both graphs have an edge that connects vertex A to vertex B, an edge that connects vertex B to vertex C, and an edge that connects vertex C to vertex D. Because the two graphs have the same number of vertices connected to each other in the same way, they are equivalent.

6. Example 2: Modeling Konigsberg with a Graph
We can represent, or model, different situations using graphs.
Example: In the early 1700’s, the city of Königsberg, Germany, was located on both banks and two islands of the Pregel River. The figure below shows that the town’s sections were connected by seven bridges.
Draw a graph that models the layout of Königsberg. Use vertices to represent the land masses and edges to represent the bridges.

7. Example 1: Modeling Konigsberg with a Graph
Solution: The only thing that matters is the relationship between land masses and bridges.
We label each of the four land masses with an uppercase letter.
We use points to represent the land masses. This figure shows vertices A, B, L, and R.
We draw two edges connecting vertex A to right bank R, one edge from A to B, and two edges from A to left bank L.

8. Example 3: Modeling Bordering Relationships for the New England States
The map of New England states are given below. Draw a graph that models which New England states share a common border. Use vertices to represent the states and edges to represent common borders.

9. Example 3: Modeling Bordering Relationships for the New England States
Solution:
Notice the states are labeled with their abbreviations.
We use the abbreviations to label each vertex and use points to represent these vertices.
Whenever two states share a common border, we connect the respective vertices with an edge.

10. Circuits & Paths Vocabulary of Graph Theory
The degree of a vertex is the number of edges at that vertex.
A vertex with an even number of edges attached to it is an even vertex.
A vertex with an odd number of edges attached to it is an odd vertex.
Two vertices in a graph are said
to be adjacent vertices if there
is at least one edge connecting
them.

11. Example 6: Identifying Adjacent Vertices
List the pairs of adjacent vertices for the given graph .
Solution: A systematic way to
approach this problem is to
first list all the pairs of adjacent
vertices involving vertex A, then
all those involving vertex B but not A,
and so on. The adjacent vertices are A and B, A and C, A and D, A and E, B and C, and E and E because there is at least one edge connecting them.

12. Circuits & Paths Vocabulary of Graph Theory
A path in a graph is a sequence of adjacent vertices and the edges connecting them.
Notice the path along this graph is represented by sequential arrows.

13. Circuits & Paths Vocabulary of Graph Theory
A circuit is a path that begins and ends at the same vertex.

14. Circuits & Paths Vocabulary of Graph Theory
A graph is connected if for any two of its vertices there is at least one path connecting them.
A graph that is not connected is called disconnected.
A disconnected graph is made of pieces that are connected by themselves called components of the graph.