Modeling with Graph Theory Presented by Dr. G.H.J. Lanel 15th May, - - PowerPoint PPT Presentation

Modeling with Graph Theory

Presented by Dr. G.H.J. Lanel 15th May, 2011

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 1 / 19

Outline

1

A Real World Problem

2

Introduction to Graph Theory

3

Definitions

4

Modeling with Graphs

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 2 / 19

A Real World Problem

Outline

1

A Real World Problem

2

Introduction to Graph Theory

3

Definitions

4

Modeling with Graphs

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 3 / 19

A Real World Problem

Example: Seven Bridges of Knigsberg

The city of Knigsberg in Russia was set on both sides of the Pregel River, and included two large islands which were connected to each

• ther and the mainland by seven bridges.

Map of Knigsberg in Euler’s time (15 April, 1707 18 September, 1783) showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 4 / 19

A Real World Problem

Example: Seven Bridges of Knigsberg

The city of Knigsberg in Russia was set on both sides of the Pregel River, and included two large islands which were connected to each

• ther and the mainland by seven bridges.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 5 / 19

A Real World Problem

Example: Seven Bridges of Knigsberg

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. Euler proved that the problem has no solution.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 6 / 19

A Real World Problem

Example: Seven Bridges of Knigsberg

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. Euler proved that the problem has no solution.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 6 / 19

A Real World Problem

Example: Seven Bridges of Knigsberg

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. Euler proved that the problem has no solution.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 6 / 19

A Real World Problem

Example: Seven Bridges of Knigsberg

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. Euler proved that the problem has no solution.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 6 / 19

A Real World Problem

Example: Seven Bridges of Knigsberg

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. Euler proved that the problem has no solution.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 6 / 19

Introduction to Graph Theory

Outline

1

A Real World Problem

2

Introduction to Graph Theory

3

Definitions

4

Modeling with Graphs

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 7 / 19

Introduction to Graph Theory

Introduction

Configurations of nodes and connections occur in a great diversity

• f applications.

They may represent physical networks, such as electrical circuits, roadways, or organic molecules. And they are also used in representing less tangible interactions as might occur in ecosystems, sociological relationships, databases, or the flow of control in a computer program. Formally, such configurations are modeled by combinotorial structures called graphs, consiisting of two sets called vertices and edges and incidence relation between them.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 8 / 19

Introduction to Graph Theory

Introduction

Configurations of nodes and connections occur in a great diversity

• f applications.

They may represent physical networks, such as electrical circuits, roadways, or organic molecules. And they are also used in representing less tangible interactions as might occur in ecosystems, sociological relationships, databases, or the flow of control in a computer program. Formally, such configurations are modeled by combinotorial structures called graphs, consiisting of two sets called vertices and edges and incidence relation between them.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 8 / 19

Introduction to Graph Theory

Introduction

Configurations of nodes and connections occur in a great diversity

• f applications.

They may represent physical networks, such as electrical circuits, roadways, or organic molecules. And they are also used in representing less tangible interactions as might occur in ecosystems, sociological relationships, databases, or the flow of control in a computer program. Formally, such configurations are modeled by combinotorial structures called graphs, consiisting of two sets called vertices and edges and incidence relation between them.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 8 / 19

Introduction to Graph Theory

Introduction

Configurations of nodes and connections occur in a great diversity

• f applications.

They may represent physical networks, such as electrical circuits, roadways, or organic molecules. And they are also used in representing less tangible interactions as might occur in ecosystems, sociological relationships, databases, or the flow of control in a computer program. Formally, such configurations are modeled by combinotorial structures called graphs, consiisting of two sets called vertices and edges and incidence relation between them.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 8 / 19

Definitions

Outline

1

A Real World Problem

2

Introduction to Graph Theory

3

Definitions

4

Modeling with Graphs

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 9 / 19

Definitions

Graph

A graph G = (V, E) is a mathematical structure consisting of two finite sets V aand E. The elements of V are called vertices (or nodes), and the elements of E are called edges.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 10 / 19

Definitions

Graph

A graph G = (V, E) is a mathematical structure consisting of two finite sets V aand E. The elements of V are called vertices (or nodes), and the elements of E are called edges.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 10 / 19

Definitions

Example: A graph with four vertices and five edges

In the graph, v1, v2, v3, v4 are vertices, and e1, e2, e3, e4, e5 are edges

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 11 / 19

Definitions

Example: A simple graph

This graph is also called complete, since each vertex has an edge to every other vertex.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 12 / 19

Definitions

Example: A simple graph

This graph is also called complete, since each vertex has an edge to every other vertex.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 12 / 19

Definitions

Example: General graphs

A graph with a loop on vertex 1.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 13 / 19

Definitions

Example: General graphs

A multigraph with multiple edges (red), and several loops (blue).

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 14 / 19

Modeling with Graphs

Outline

1

A Real World Problem

2

Introduction to Graph Theory

3

Definitions

4

Modeling with Graphs

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 15 / 19

Modeling with Graphs

Example: Seven Bridges of Knigsberg

The city of Knigsberg in Russia was set on both sides of the Pregel River, and included two large islands which were connected to each

• ther and the mainland by seven bridges.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 16 / 19

Modeling with Graphs

Example: Seven Bridges of Knigsberg

In abstract terms, one replaces each land mass with an abstract ”vertex” or node, and each bridge with an abstract connection, an ”edge”, which only serves to record which pair of vertices (land masses) is connected by that bridge. The resulting mathematical structure is called a graph.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 17 / 19

Modeling with Graphs

Example: Seven Bridges of Knigsberg

Euler shows that the possibility of a walk through a graph, traversing each edge exactly once, depends on the degrees of the

• nodes. The degree of a node is the number of edges touching it.

Euler’s argument shows that a necessary condition for the walk (Euler tour) of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 18 / 19

Modeling with Graphs

Example: Seven Bridges of Knigsberg

Euler shows that the possibility of a walk through a graph, traversing each edge exactly once, depends on the degrees of the

• nodes. The degree of a node is the number of edges touching it.

Euler’s argument shows that a necessary condition for the walk (Euler tour) of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 18 / 19

Modeling with Graphs

Example: Seven Bridges of Knigsberg

Every vertex of this graph has an even degree, therefore this is an Euler tour. Following the edges in alphabetical order gives an Euler tour.

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 19 / 19

Modeling with Graphs

Thank You!

Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 20 / 19