1. Discrete Structures – CNS2300
Text
Discrete Mathematics and Its Applications (5th Edition)
Kenneth H. Rosen
Chapter 8
Graphs

2. Section 8.4
Connectivity

3. Paths
A path is a sequence of edges that begins at a vertex of a graph and travels along edges of the graph, always connecting pairs of adjacent vertices.
The path is a circuit if it begins and ends at the same vertex.
The path or circuit is said to pass through the vertices or traverse the edges
A path or circuit is simple if it does not contain the same edge more than once.

4. Paths
a,b ,d ,g ,f e g a b d f c

5. Circuits, Simple Path or Circuitb c d e f g

6. Paths in Directed Graphsb c d e f

7. Acquaintanceship Graphs
Bacon No.
No. People

8. Counting Paths Between Vertices
Let G be a graph with adjacency matrix A. The number of different paths of length r from vi to vj, where r is a positive integer, equals the (i, j)th entry of Ar

9. Connectedness Connected Undirected Connected Directed
Simple path between every pair of distinct vertices
Connected Directed
Strongly Connected
Weakly Connected

10. Euler & Hamilton Paths
Bridges of Konigsberg

11. Euler Circuit
An Euler circuit in a graph G is a simple circuit containing every edge of G.
An Euler path in G is a simple path containing every edge of G.

12. Necessary & Sufficient Conditions
A connected multigraph has an Euler circuit if and only if each of its vertices has even degree
A connected multigraph has an Euler path but not an Euler circuit if and only if it has exactly two vertices of odd degree.

13. Hamilton Paths and Circuits
A Hamilton circuit in a graph G is a simple circuit passing through every vertex of G, exactly once.
An Hamilton Path in G is a simple path passing through every vertex of G, exactly once.

14. Conditions
If G is a simple graph with n vertices n>=3 such that the degree of every vertex in G is at least n/2, then G has a Hamilton circuit.
If G is a simple graph with n vertices n>=3 such that deg(u)+deg(v)>=n for every pair of nonadjacent vertices u and v in G, then G has a Hamilton circuit.

15. finished