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 A Real World Problem 1 Introduction to Graph Theory 2 Definitions 3 Modeling with Graphs 4 Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 2 / 19
A Real World Problem Outline A Real World Problem 1 Introduction to Graph Theory 2 Definitions 3 Modeling with Graphs 4 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 other 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 other 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 A Real World Problem 1 Introduction to Graph Theory 2 Definitions 3 Modeling with Graphs 4 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 of 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 of 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 of 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 of 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 A Real World Problem 1 Introduction to Graph Theory 2 Definitions 3 Modeling with Graphs 4 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, v 1 , v 2 , v 3 , v 4 are vertices, and e 1 , e 2 , e 3 , e 4 , e 5 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 A Real World Problem 1 Introduction to Graph Theory 2 Definitions 3 Modeling with Graphs 4 Presented by Dr. G.H.J. Lanel (USJP) Modeling with Graph Theory 15th May, 2011 15 / 19
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