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Discrete Structures – CNS2300

Text

Discrete Mathematics and Its Applications (5th Edition)

Kenneth H. Rosen

Chapter 8

Graphs

Section 8.4

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.

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.

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