# 数学代写|图论代写Graph Theory代写|MATH-3020

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## 数学代写|图论代写Graph Theory代写|Homogeneous and universal graphs

Unlike finite graphs, infinite graphs offer the possibility to represent an entire graph property $\mathcal{P}$ by just one specimen, a single graph that contains all the graphs in $\mathcal{P}$ up to some fixed cardinality. Such graphs are called ‘universal’ for this property.

More precisely, if $\leqslant$ is a graph relation (such as the minor, topological minor, subgraph, or induced subgraph relation up to isomorphism), we call a countable graph $G^$ universal in $\mathcal{P}$ (for $\leqslant$ ) if $G^ \in \mathcal{P}$ and $G \leqslant G^*$ for every countable graph $G \in \mathcal{P}$.

Is there a graph that is universal in the class of all countable graphs? Suppose a graph $R$ has the following property:
Whenever $U$ and $W$ are disjoint finite sets of vertices in $R$, there exists a vertex $v \in R-U-W$ that is adjacent in $R$ to all the vertices in $U$ but to none in $W$.
Then $R$ is universal even for the strongest of all graph relations, the induced subgraph relation. Indeed, in order to embed a given countable graph $G$ in $R$ we just map its vertices $v_1, v_2, \ldots$ to $R$ inductively, making sure that $v_n$ gets mapped to a vertex $v \in R$ adjacent to the images of all the neighbours of $v_n$ in $G\left[v_1, \ldots, v_n\right]$ but not adjacent to the image of any non-neighbour of $v_n$ in $G\left[v_1, \ldots, v_n\right]$. Clearly, this map is an isomorphism between $G$ and the subgraph of $R$ induced by its image.
Theorem 8.3.1. (Erdős and Rényi 1963)
There exists a unique countable graph $R$ with property (). Proof. To prove existence, we construct a graph $R$ with property () inductively. Let $R_0:=K^1$. For all $n \in \mathbb{N}$, let $R_{n+1}$ be obtained from $R_n$ by adding for every set $U \subseteq V\left(R_n\right)$ a new vertex $v$ joined to all the vertices in $U$ but to none outside $U$. (In particular, the new vertices form an independent set in $R_{n+1}$.) Clearly $R:=\bigcup_{n \in \mathbb{N}} R_n$ has property (). To prove uniqueness, let $R=(V, E)$ and $R^{\prime}=\left(V^{\prime}, E^{\prime}\right)$ be two graphs with property $()$, each given with a fixed vertex enumeration. We construct a bijection $\varphi: V \rightarrow V^{\prime}$ in an infinite sequence of steps, defining $\varphi(v)$ for one new vertex $v \in V$ at each step.

## 数学代写|图论代写Graph Theory代写|Connectivity and matching

In this section we look at infinite versions of Menger’s theorem and of the matching theorems from Chapter 2. This area of infinite graph theory is one of its best developed fields, with several deep results. One of these, however, stands out among the rest: a version of Menger’s theorem that had been conjectured by Erdős and was proved only recently by Aharoni and Berger. The techniques developed for its proof inspired, over the years, much of the theory in this area.

We shall prove this theorem for countable graphs, which will take up most of this section. Although the countable case is much easier, it is still quite hard and will give a good impression of the general proof. We then wind up with an overview of infinite matching theorems and a conjecture conceived in the same spirit.

Recall that Menger’s theorem, in its simplest form, says that if $A$ and $B$ are sets of vertices in a finite graph $G$, not necessarily disjoint, and if $k=k(G, A, B)$ is the minimum number of vertices separating $A$ from $B$ in $G$, then $G$ contains $k$ disjoint $A-B$ paths. (Clearly, it cannot contain more.) The same holds, and is easily deduced from the finite case, when $G$ is infinite but $k$ is still finite:

Proposition 8.4.1. Let $G$ be any graph, $k \in \mathbb{N}$, and let $A, B$ be two sets of vertices in $G$ that can be separated by $k$ but no fewer than $k$ vertices. Then $G$ contains $k$ disjoint $A-B$ paths.

Proof. By assumption, every set of disjoint $A-B$ paths has cardinality at most $k$. Choose one, $\mathcal{P}$ say, of maximum cardinality. Suppose $|\mathcal{P}|<k$. Then no set $X$ consisting of one vertex from each path in $\mathcal{P}$ separates $A$ from $B$. For each $X$, let $P_X$ be an $A-B$ path avoiding $X$. Let $H$ be the union of $\bigcup \mathcal{P}$ with all these paths $P_X$. This is a finite graph in which no set of $|\mathcal{P}|$ vertices separates $A$ from $B$. So $H \subseteq G$ contains more than $|\mathcal{P}|$ paths from $A$ to $B$ by Menger’s theorem (3.3.1), which contradicts the choice of $\mathcal{P}$.

When $k$ is infinite, however, the result suddenly becomes trivial. Indeed, let $\mathcal{P}$ be any maximal set of disjoint $A-B$ paths in $G$. Then the union of all these paths separates $A$ from $B$, so $\mathcal{P}$ must be infinite. But then the cardinality of this union is no bigger than $|\mathcal{P}|$. Thus, $\mathcal{P}$ contains $|\mathcal{P}|=|\bigcup \mathcal{P}| \geqslant k$ disjoint $A-B$ paths, as desired.

## 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。