Graph Theory
V. Adamchik
1
Graph Theory
Victor Adamchik Fall of 2005
Plan
1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs
Introduction
A.Aho and J.Ulman acknowledge that “Fundamentally, computer science is a science of abstraction.” Computer scientists must create abstractions of real-world problems that can be represented and manipulated in a computer. Sometimes the process of abstraction is simple. For example, we use a logic to design a computer circuits. Another example - scheduling final exams. For successful scheduling we have to take into account associations between courses, students and rooms. Such set of connections between items is modeled by graphs. Let me reiterate, in our model the set of items (courses, students and rooms) won't be much helpful. We also have to have a set of connections between pairs of items, because we need to study the relationships between connections. The basic idea of graphs were introduced in 18th century by the great Swiss mathematician Leonhard Euler. He used graphs to solve the famous Königsberg bridge problem. Here is a picture (taken from the internet)
V. Adamchik
21-127: Concepts of Mathematics
German city of Königsberg (now it is Russian Kaliningrad) was situated on the river Pregel. It had a park situated on the banks of the river and two islands. Mainland and islands were joined by seven bridges. A problem was whether it was possible to take a walk through the town in such a way as to cross over every bridge once, and only once. Here is the graph model of the problem
A graph is a set of points (we call them vertices or nodes) connected by lines (edges or arcs). The simplest example known to you is a linked list.
V. Adamchik
3
The Web: The entire Web is a graph, where items are documents and the references (links) are connections. Networks: A network consist of sites that send and recieve messages of various types. Program Structure: A compiler builds a graph to represent relationships
1
Graph Theory
Victor Adamchik Fall of 2005
Plan
1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs
Introduction
A.Aho and J.Ulman acknowledge that “Fundamentally, computer science is a science of abstraction.” Computer scientists must create abstractions of real-world problems that can be represented and manipulated in a computer. Sometimes the process of abstraction is simple. For example, we use a logic to design a computer circuits. Another example - scheduling final exams. For successful scheduling we have to take into account associations between courses, students and rooms. Such set of connections between items is modeled by graphs. Let me reiterate, in our model the set of items (courses, students and rooms) won't be much helpful. We also have to have a set of connections between pairs of items, because we need to study the relationships between connections. The basic idea of graphs were introduced in 18th century by the great Swiss mathematician Leonhard Euler. He used graphs to solve the famous Königsberg bridge problem. Here is a picture (taken from the internet)
V. Adamchik
21-127: Concepts of Mathematics
German city of Königsberg (now it is Russian Kaliningrad) was situated on the river Pregel. It had a park situated on the banks of the river and two islands. Mainland and islands were joined by seven bridges. A problem was whether it was possible to take a walk through the town in such a way as to cross over every bridge once, and only once. Here is the graph model of the problem
A graph is a set of points (we call them vertices or nodes) connected by lines (edges or arcs). The simplest example known to you is a linked list.
V. Adamchik
3
The Web: The entire Web is a graph, where items are documents and the references (links) are connections. Networks: A network consist of sites that send and recieve messages of various types. Program Structure: A compiler builds a graph to represent relationships