1. Lecture 15: Graph Theory II
Discrete Mathematical Structures:
Theory and Applications

2. Learning Objectives Learn the basic properties of graph theory
Learn about walks, trails, paths, circuits, and cycles in a graph
Explore how graphs are represented in computer memory
Learn about Euler and Hamilton circuits
Learn about isomorphism of graphs
Explore various graph algorithms
Examine planar graphs and graph coloring
Discrete Mathematical Structures: Theory and Applications

3. Matrix Representation of a Graph
Discrete Mathematical Structures: Theory and Applications

4. Matrix Representation of a Graph
Discrete Mathematical Structures: Theory and Applications

5. Matrix Representation of a Graph
Discrete Mathematical Structures: Theory and Applications

6. Discrete Mathematical Structures: Theory and Applications

7. Discrete Mathematical Structures: Theory and Applications

8. Matrix Representation of a Graph
Discrete Mathematical Structures: Theory and Applications

9. Matrix Representation of a Graph
Discrete Mathematical Structures: Theory and Applications

10. Matrix Representation of a Graph
Discrete Mathematical Structures: Theory and Applications

11. Special Circuits
Discrete Mathematical Structures: Theory and Applications

12. Special Circuits
Discrete Mathematical Structures: Theory and Applications

13. Special Circuits
Discrete Mathematical Structures: Theory and Applications

14. Special Circuits
Discrete Mathematical Structures: Theory and Applications

15. Special Circuits Definition 10.4.9 Example 10.4.10
An open trail in a graph is called an Euler trail if it contains all the edges and all the vertices.
Example
This is a connected graph. It has a vertex of odd degree. Thus, this graph has no Euler circuit, but the trail (v2, e7, v6, e5, v5, e4, v4, e3, v3, e2, v2, e1, v1, e6, v6) contains all the edges of G. Hence, this is an Euler trail.
Discrete Mathematical Structures: Theory and Applications

16. Special Circuits
Discrete Mathematical Structures: Theory and Applications

17. Special Circuits
This diagram in Figure is a connected graph with 20 vertices. Each Vertex represents a famous city. It follows that the game is equivalent to finding a cycle in the graph in Figure that contains each vertex exactly once except for the starting and ending vertices, which appear twice, making a Hamiltonian Cycle.
Discrete Mathematical Structures: Theory and Applications

18. Special Circuits
Discrete Mathematical Structures: Theory and Applications

19. Special Circuits
Discrete Mathematical Structures: Theory and Applications

20. Special Circuits
Discrete Mathematical Structures: Theory and Applications

21. Special Circuits
During a certain soccer tournament with 4 teams, each team has played against all the others exactly once and there were no ties. All the teams can be listed in order so that each has defeated the team next on the list.
Let the teams be denoted by v1, v2, v3, and v4 and let the matches correspond to the vertices and the arcs of a directed graph, respectively, in such a way that the initial and terminal vertices of an arc correspond to the winner and loser, respectively, of the corresponding match.
v2 → v1 → v3 → v4 is a Hamiltonian directed path.
Discrete Mathematical Structures: Theory and Applications