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# 数学代写|组合学代写Combinatorics代考|What is a Graph?

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## 数学代写|组合学代写Combinatorics代考|What is a Graph?

There are various types of graphs, each with its own definition. Unfortunately, some people apply the term “graph” rather loosely, so you can’t be sure what type of graph they’re talking about unless you ask them. After you have finished this chapter, we expect you to use the terminology carefully, not loosely. To motivate the various definitions, we’ll begin with some examples.

Example 5.1 A computer network Computers are often linked with one another so that they can interchange information. Given a collection of computers, we would like to describe this linkage in fairly clean terms so that we can answer questions such as “How can we send a message from computer A to computer B using the fewest possible intermediate computers?”

We could do this by making a list that consists of pairs of computers that are connected. Note that these pairs are unordered since, if computer $\mathrm{C}$ can communicate with computer D, then the reverse is also true. (There are sometimes exceptions to this, but they are rare and we will assume that our collection of computers does not have such an exception.) Also, note that we have implicitly assumed that the computers are distinguished from each other: It is insufficient to say that “A PC is connected to a Mac.” We must specify which PC and which Mac. Thus, each computer has a unique identifying label of some sort.

For people who like pictures rather than lists, we can put dots on a piece of paper, one for each computer. We label each dot with a computer’s identifying label and draw a curve connecting two dots if and only if the corresponding computers are connected. Note that the shape of the curve does not matter (it could be a straight line or something more complicated) because we are only interested in whether two computers are connected or not. Figure 5.1 shows such a picture. Each computer has been labeled by the initials of its owner.

Recall that $\mathcal{P}_2(V)$ stands for the set of all two element subsets of the set $V$. Based on our computer example we have

Definition 5.1 Simple graph A simple graph $G$ is a set $V$, called the vertices of $G$, and a subset $E$ of $\mathcal{P}_2(V)$ (i.e., a set $E$ of 2 element subsets of $V$ ), called the edges of $G$. We can represent this by writing $G=(V, E)$.

In our case, the vertices are the computers and a pair of computers is in $E$ if and only if they are connected.

## 数学代写|组合学代写Combinatorics代考|Equivalence Relations and Unlabeled Graphs

Sometimes we are interested only in the “structure” of a graph and not in the names (labels) of the vertices and edges. In this case we are interested in what is called an unlabeled graph. A picture of an unlabeled graph can be obtained from a picture of a graph by erasing all of the names on the vertices and edges. This concept is simple enough, but is difficult to use mathematically because the idea of a picture is not very precise.

The concept of an equivalence relation on a set is an important concept in mathematics and computer science. We used the idea in Section 4.3, but did not discuss it much there. We’ll explore it more fully here and will use it to rigorously define unlabeled graphs. Later we will use it to define connected components and biconnected components. We recall the definition given in Section 4.3:

Definition 5.3 Equivalence relation An equivalence relation on a set $S$ is a partition of $S$. We say that $s, t \in S$ are equivalent if and only if they belong to the same block. If the symbol $\sim$ denotes the equivalence relation, then we write $s \sim t$ to indicate that $s$ and $t$ are equivalent.
Example 5.4 To refresh your memory, we’ll look at some simple equivalence relations.
Let $S$ be any set and let all the blocks of the partition have one element. Two elements of $S$ are equivalent if and only if they are the same. This rather trivial equivalence relation is, of course, denoted by “=”.

Now let the set be the integers $\mathbb{Z}$. Let’s try to define an equivalence relation by saying that $n$ and $k$ are equivalent if and only if they differ by a multiple of 24 . Is this an equivalence relation? If it is we should be able to find the blocks of the partition. There are 24 of them, which we could number $0, \ldots, 23$. Block $j$ consists of all integers which equal $j$ plus a multiple of 24 ; that is, they have a remainder of $j$ when divided by 24 . Since two numbers belong to the same block if and only if they both have the same remainder when divided by 24 , it follows that they belong to the same block if and only if their difference gives a remainder of 0 when divided by 24 , which is the same as saying their difference is a multiple of 24 . Thus this partition does indeed give the desired equivalence relation.

## 数学代写|组合学代写Combinatorics代考|Equivalence Relations and Unlabeled Graphs

5.3等价关系集合$S$上的等价关系是$S$的一个分区。我们说$s, t \在s$中是等价的当且仅当它们属于同一块。如果符号$\sim$表示等价关系，那么我们写$s \sim t$表示$s$和$t$是等价的。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。