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Discrete Structures – CNS2300. Text Discrete Mathematics and Its Applications (5 th Edition) Kenneth H. Rosen Chapter 8 Graphs. Section 8.4. Connectivity. Paths.

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Discrete Structures – CNS2300

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### Discrete Structures – CNS2300

Text

Discrete Mathematics and Its Applications (5th Edition)

Kenneth H. Rosen

Chapter 8

Graphs

Connectivity

### 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.

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### Acquaintanceship Graphs

http://www.cs.virginia.edu/oracle/

http://www.brunching.com/bacondegrees.html

### 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

### Connectedness

• Connected Undirected

• Simple path between every pair of distinct vertices

• Connected Directed

• Strongly Connected

• Weakly Connected

### Euler & Hamilton Paths

Bridges ofKonigsberg

### 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.

### 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.

### 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.

### 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.