Download presentation

Published byPatricia Lambert Modified over 7 years ago

1
**Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley.**

All rights reserved

2
**© 2008 Pearson Addison-Wesley. All rights reserved**

Chapter 15: Graph Theory 15.1 Basic Concepts 15.2 Euler Circuits 15.3 Hamilton Circuits and Algorithms 15.4 Trees and Minimum Spanning Trees © 2008 Pearson Addison-Wesley. All rights reserved

3
**© 2008 Pearson Addison-Wesley. All rights reserved**

Chapter 1 Section 15-1 Basic Concepts © 2008 Pearson Addison-Wesley. All rights reserved

4
**© 2008 Pearson Addison-Wesley. All rights reserved**

Basic Concepts Graphs Walks, Paths, and Circuits Complete Graphs and Subgraphs Graph Coloring © 2008 Pearson Addison-Wesley. All rights reserved

5
**© 2008 Pearson Addison-Wesley. All rights reserved**

Simple Graphs Graphs in which there is no more than one edge between any two vertices and in which no edge goes from vertex to the same vertex are called simple. Simple Not simple © 2008 Pearson Addison-Wesley. All rights reserved

6
**© 2008 Pearson Addison-Wesley. All rights reserved**

Degree of the Vertex The number of edges joined to a vertex is called the degree of the vertex. C A B D A has degree 0, B has degree 1, C has degree 2 and D has degree 3. © 2008 Pearson Addison-Wesley. All rights reserved

7
**© 2008 Pearson Addison-Wesley. All rights reserved**

Isomorphic Graphs Two graphs are isomorphic if there is a one-to-one matching between vertices of the two graphs with the property that whenever there is an edge between two vertices of either one of the graphs, there is an edge between the corresponding vertices of the other graph. © 2008 Pearson Addison-Wesley. All rights reserved

8
**Connected and Disconnected Graphs**

A graph is connected if one can move from each vertex of the graph to every other vertex of the graph along edges of the graph. If not, the graph is disconnected. The connected pieces of a graph are called the components of the graph. © 2008 Pearson Addison-Wesley. All rights reserved

9
**Example: Isomorphic Graphs**

Are the two graphs isomorphic? Solution Yes, and corresponding vertices are labeled below. D A B D B C A C © 2008 Pearson Addison-Wesley. All rights reserved

10
**© 2008 Pearson Addison-Wesley. All rights reserved**

Sum of Degrees Theorem In any graph, the sum of the degrees of the vertices equals twice the number of edges. © 2008 Pearson Addison-Wesley. All rights reserved

11
**Example: Sum of Degrees**

A graph has exactly four vertices, each of degree 3. How many edges does this graph have? Solution The sum of degrees of the vertices is 12. By the theorem, this number is twice the number of edges, so the number of edges is 12/2 = 6. © 2008 Pearson Addison-Wesley. All rights reserved

12
**© 2008 Pearson Addison-Wesley. All rights reserved**

Walk A walk in a graph is a sequence of vertices, each linked to the next vertex by a specified edge of the graph. © 2008 Pearson Addison-Wesley. All rights reserved

13
**© 2008 Pearson Addison-Wesley. All rights reserved**

Path A path in a graph is a walk that uses no edge more than once. © 2008 Pearson Addison-Wesley. All rights reserved

14
**© 2008 Pearson Addison-Wesley. All rights reserved**

Circuit A circuit in a graph is a path that begins and ends at the same vertex. © 2008 Pearson Addison-Wesley. All rights reserved

15
**Venn Diagram of Walks, Paths, and Circuits**

© 2008 Pearson Addison-Wesley. All rights reserved

16
**Example: Classifying Walks**

Using the graph, classify each sequence as a walk, a path or a circuit. A E B D C © 2008 Pearson Addison-Wesley. All rights reserved

17
**Example: Classifying Walks**

Solution B D C Walk Path Circuit a) No No b) Yes Yes c) Yes d) Yes © 2008 Pearson Addison-Wesley. All rights reserved

18
**© 2008 Pearson Addison-Wesley. All rights reserved**

Complete Graph A complete graph is a graph in which there is exactly one edge going from each vertex to each other vertex in the graph. © 2008 Pearson Addison-Wesley. All rights reserved

19
**Example: Complete Graph**

Draw a complete graph with four vertices. Solution Answers may vary, but one solution is shown below. © 2008 Pearson Addison-Wesley. All rights reserved

20
**© 2008 Pearson Addison-Wesley. All rights reserved**

Subgraph A graph consisting of some of the vertices of the original graph and some the original edges between those vertices is called a subgraph. © 2008 Pearson Addison-Wesley. All rights reserved

21
**Coloring and Chromatic Number**

A coloring for a graph is a coloring of the vertices in such a way that the vertices joined by an edge have different colors. The chromatic number of a graph is the least number of colors needed to make a coloring. © 2008 Pearson Addison-Wesley. All rights reserved

22
**© 2008 Pearson Addison-Wesley. All rights reserved**

Coloring a Graph Step 1: Choose a vertex with highest degree, and color it. Use the same color to color as many vertices as you can without coloring vertices joined by an edge of the same color. Step 2: Choose a new color, and repeat what you did in Step 1 for vertices not already colored. Step 3: Repeat Step 1 until all vertices are colored. © 2008 Pearson Addison-Wesley. All rights reserved

23
**Example: Coloring a Graph**

Color the graph below and give its chromatic number . Solution Its chromatic number is 3. © 2008 Pearson Addison-Wesley. All rights reserved

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google