1. Discrete Mathematics Lecture 13_14: Graph Theory and Tree
By:
Nur Uddin, Ph.D

2. Definition of graph

3. Basic terminology

4. The Handshaking Theorem
Example:

5. Complete Graphs

6. Isomorphic
Sometimes, two graphs have exactly the same form, in the sense that there is a one-to-one correspondence between their vertex sets that preserves edges. In such a case, we say that the two graphs are isomorphic.
One way to represent a graph without multiple edges is to list all the edges of this graph. Another way to represent a graph with no multiple edges is to use adjacency lists, which specify the vertices that are adjacent to each vertex of the graph.

7. Example: Adjecency list

8. Adjacency matrix

9. Incidence Matrix

10. Isomorphic

11. Isomorphic
Isomorphic simple graphs also must have the same number of edges, because the one-to-one correspondence between vertices establishes a one-to-one correspondence between edges.

12. Isomorphic evaluation using matrix

13. Connectivity
A path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph.

14. Example

15. Connected components

16. Euler and Hamilton Paths

17. Theorems

18. Example

19. Hamilton Paths and Circuits

20. Hamilton Puzzle

21. Example

22. Shortest path problem

23. Exercise
Determine the shortest distance form a to z.

24. Dijkstra’s Algorithm