Презентация Graph theory irina prosvirnina. Definitions and examples. Paths and cycles онлайн
На нашем сайте вы можете скачать и просмотреть онлайн доклад-презентацию на тему Graph theory irina prosvirnina. Definitions and examples. Paths and cycles абсолютно бесплатно. Урок-презентация на эту тему содержит всего 68 слайдов. Все материалы созданы в программе PowerPoint и имеют формат ppt или же pptx. Материалы и темы для презентаций взяты из открытых источников и загружены их авторами, за качество и достоверность информации в них администрация сайта не отвечает, все права принадлежат их создателям. Если вы нашли то, что искали, отблагодарите авторов - поделитесь ссылкой в социальных сетях, а наш сайт добавьте в закладки.
Презентации » Математика » Graph theory irina prosvirnina. Definitions and examples. Paths and cycles
Оцените!
Оцените презентацию от 1 до 5 баллов!
- Тип файла:ppt / pptx (powerpoint)
- Всего слайдов:68 слайдов
- Для класса:1,2,3,4,5,6,7,8,9,10,11
- Размер файла:3.38 MB
- Просмотров:72
- Скачиваний:0
- Автор:неизвестен
Слайды и текст к этой презентации:
№2 слайд
![Definitions and examples](/documents_6/837100c75af440d859ddcc77f0ee9f21/img1.jpg)
Содержание слайда: Definitions and examples
Although generally regarded as one of the more modern branches of mathematics, graph theory actually dates back to 1736.
In that year Leonhard Euler published the first paper on what is now called graph theory. In the paper, Euler developed a theory which solved the so-called Knigsberg Bridge problem.
№5 слайд
![Definitions and examples What](/documents_6/837100c75af440d859ddcc77f0ee9f21/img4.jpg)
Содержание слайда: Definitions and examples
What is a ‘graph’? Intuitively, a graph is simply a collection of points, called ‘vertices’, and a collection of lines, called ‘edges’, each of which joins either a pair of points or a single point to itself.
A familiar example, which serves as a useful analogy, is a road map which shows towns as vertices and the roads joining them as edges.
№6 слайд
![Definitions and examples](/documents_6/837100c75af440d859ddcc77f0ee9f21/img5.jpg)
Содержание слайда: Definitions and examples
Definition 1
An undirected graph comprises:
a finite non-empty set of vertices,
a finite set E of edges, and
a function : E such that, for every edge , is a one- or two-element subset of .
The edge e is said to join the element(s) of .
An undirected graph is simple if there are no loops and each pair of distinct vertices is joined by a unique edge.
№8 слайд
![Definitions and examples We](/documents_6/837100c75af440d859ddcc77f0ee9f21/img7.jpg)
Содержание слайда: Definitions and examples
We should emphasize that a graph and a diagram representing it are not the same thing.
A given graph may be represented by two diagrams which appear very different.
For instance, the two diagrams in the figure represent the same graph as can be observed by writing down the function
: E
№10 слайд
![Definitions and examples](/documents_6/837100c75af440d859ddcc77f0ee9f21/img9.jpg)
Содержание слайда: Definitions and examples
Definition 2
The degree or valency, deg(), of a vertex is the number of edges which are incident to . (Unless stated otherwise, a loop joining to itself counts two towards the degree of .)
A graph in which every vertex has the same degree is called regular (with degree ) or simply -regular.
№17 слайд
![Definitions and examples](/documents_6/837100c75af440d859ddcc77f0ee9f21/img16.jpg)
Содержание слайда: Definitions and examples
Definition 3
A null graph (or totally disconnected graph) is one whose edge set is empty. (Pictorially, a null graph is just a collection of points.)
A complete graph is a simple graph in which every pair of distinct vertices is joined by an edge.
A bipartite graph is a graph where the vertex set has a partition such that every edge joins a vertex of to a vertex of .
A complete bipartite graph is a bipartite graph such that every vertex of is joined to every vertex of by a unique edge.
№19 слайд
![Definitions and examples](/documents_6/837100c75af440d859ddcc77f0ee9f21/img18.jpg)
Содержание слайда: Definitions and examples
Example 1
The complete graph with vertices can be described as follows.
It has vertex set and edge set with the function given by .
The graph is clearly regular with degree , since every vertex is connected, by a unique edge, to each of the other vertices.
№21 слайд
![Definitions and examples](/documents_6/837100c75af440d859ddcc77f0ee9f21/img20.jpg)
Содержание слайда: Definitions and examples
Example 2
Let be a bipartite graph where the vertex set has the partition .
Note that need not be simple. All that is required is that each edge must join a vertex of to a vertex of . Given and , there may be more than one edge joining them or no edge joining them.
Clearly, though, there are no loops in .
№25 слайд
![Definitions and examples The](/documents_6/837100c75af440d859ddcc77f0ee9f21/img24.jpg)
Содержание слайда: Definitions and examples
The adjacency matrix is necessarily symmetric as the number of edges joining and is the same as the number joining and .
The degree of vertex is easily determined from the adjacency matrix.
If there are no loops at then its degree is the sum of the entries in the th column (or th row) of the matrix.
Since every loop counts twice in the degree, when summing the entries in the th column (or th row) the diagonal element must be doubled to obtain the degree of .
№33 слайд
![Definitions and examples](/documents_6/837100c75af440d859ddcc77f0ee9f21/img32.jpg)
Содержание слайда: Definitions and examples
Definition 5
A graph is a subgraph of the graph , denoted , if , and , for every edge of .
The condition that , for every edge of , just means that the edges of the subgraph must join the same vertices as they do in . Intuitively, is a subgraph of if we can obtain a diagram for by erasing some of the vertices and/or edges from a diagram of . Of course, if we erase a vertex we must also erase all edges incident to it.
№35 слайд
![Paths and cycles Using the](/documents_6/837100c75af440d859ddcc77f0ee9f21/img34.jpg)
Содержание слайда: Paths and cycles
Using the analogy of a road map, we can consider various types of ‘journeys’ in a graph.
For instance, if the graph actually represents a network of roads connecting various towns, one question we might ask is: is there a journey, beginning and ending at the same town, which visits every other town just once without traversing the same road more than once?
As usual, we begin with some definitions.
№36 слайд
![Paths and cycles Definition](/documents_6/837100c75af440d859ddcc77f0ee9f21/img35.jpg)
Содержание слайда: Paths and cycles
Definition 6
An edge sequence of length in a graph is a sequence of (not necessarily distinct) edges such that and are adjacent for . The edge sequence determines a sequence of vertices (again, not necessarily distinct) where (). We say is the initial vertex and the final vertex of the edge sequence.
№54 слайд
![? In the -entry represents](/documents_6/837100c75af440d859ddcc77f0ee9f21/img53.jpg)
Содержание слайда: ? In the -entry represents the number of edge sequences of length joining and .
The - entry of is obtained by ‘multiplying’ the th row and the th column of . We express this as
.
The th term in this sum, is the product of the number of edges connecting and with the number of edges connecting and ; in other words the number of edge sequences of length 2 joining and via .
Summing over all gives the total number of length 2 edge sequences connecting and .
№62 слайд
![Paths and cycles This means](/documents_6/837100c75af440d859ddcc77f0ee9f21/img61.jpg)
Содержание слайда: Paths and cycles
This means that is a component of if it is a connected subgraph of and it is not itself a proper subgraph of any other connected subgraph of .
This second condition is what we mean by the term maximal; it says that if is a connected subgraph such that , then so there is no connected subgraph of which is ‘bigger’ than .
№63 слайд
![Paths and cycles The](/documents_6/837100c75af440d859ddcc77f0ee9f21/img62.jpg)
Содержание слайда: Paths and cycles
The components of a graph are just its connected ‘pieces’.
In particular, a connected graph has only one component.
Decomposing a graph into its components can be very useful.
It is usually simpler to prove results about connected graphs and properties of arbitrary graphs can frequently then be deduced by considering each component in turn.
№65 слайд
![Paths and cycles if and only](/documents_6/837100c75af440d859ddcc77f0ee9f21/img64.jpg)
Содержание слайда: Paths and cycles
if and only if and can be joined by a path in .
? is an equivalence relation.
The only difficulty is in proving transitivity.
If is a path from to and is a path from to , then the edge sequence ‘ followed by ’ is an edge sequence from to , but it may not be a path as and may have edges in common.
If this is the case the edge sequence needs to be modified by omitting some edges to give the required path from to
№66 слайд
![Paths and cycles if and only](/documents_6/837100c75af440d859ddcc77f0ee9f21/img65.jpg)
Содержание слайда: Paths and cycles
if and only if and can be joined by a path in .
Let be the partition of the vertex set by the equivalence classes of .
We can now form subgraphs with vertex and whose edges are those of which joint two vertices of .
These subgraphs are the components of .
№68 слайд
![Paths and cycles Example](/documents_6/837100c75af440d859ddcc77f0ee9f21/img67.jpg)
Содержание слайда: Paths and cycles
Example 7
Frequently it is clear from a diagram of how many components it has. Sometimes, however, we need to examine the diagram more closely. For instance, both graphs illustrated in the figure have two components, although this is not instantly apparent for the graph (b).
Скачать все slide презентации Graph theory irina prosvirnina. Definitions and examples. Paths and cycles одним архивом:
Похожие презентации
-
Functions and their graphs
-
Graphs and Multigraphs
-
Graph theory
-
Recall the definitions of position, distance, displacement, velocity and acceleration
-
Descriptive statistics. Frequency distributions and their graphs. (Section 2. 1)
-
Chapter 1. Polynomial and Rational Functions. 3. 3. Dividing Polynomials; Remainder and Factor Theorems
-
Chapter 3. Polynomial and Rational Functions. 3. 2 Polynomial Functions and Their Graphs
-
Functions and graphs. Chapter 2. Combinations of functions; composite functions
-
Basics of functions and their graphs
-
Cеребренникова Ирина Павловна учитель математики МОУ «Вырицкая СОШ 1» Стаж работы 25 лет Вторая категория E-mail iserebro47yandex. ru