# 数学代写|图论代写Graph Theory代写|Math530

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## 数学代写|图论代写Graph Theory代写|Basic notions, facts and techniques

This section gives a gentle introduction to the aspects of infinity most commonly encountered in graph theory. ${ }^2$

After just a couple of definitions, we begin by looking at a few obvious properties of infinite sets, and how they can be employed in the context of graphs. We then illustrate how to use the three most basic common tools in infinite graph theory: Zorn’s lemma, transfinite induction, and something called ‘compactness’. We complete the section with the combinatorial definition of an end; topological aspects will be treated in Section 8.5.

A graph is locally finite if all its vertices have finite degrees. An infinite graph $(V, E)$ of the form
$$V=\left{x_0, x_1, x_2, \ldots\right} \quad E=\left{x_0 x_1, x_1 x_2, x_2 x_3, \ldots\right}$$
is called a ray, and a double ray is an infinite graph $(V, E)$ of the form
$$V=\left{\ldots, x_{-1}, x_0, x_1, \ldots\right} \quad E=\left{\ldots, x_{-1} x_0, x_0 x_1, x_1 x_2, \ldots\right}$$
in both cases the $x_n$ are assumed to be distinct. Thus, up to isomorphism, there is only one ray and one double ray, the latter being the unique infinite 2-regular connected graph. In the context of infinite graphs, finite paths rays and double rays are all called paths.

The subrays of a ray or double ray are its tails. Formally, every ray has infinitely many tails, but any two of them differ only by a finite initial segment. The union of a ray $R$ with infinitely many disjoint finite paths having precisely their first vertex on $R$ is a comb; the last vertices of those paths are the teeth of this comb, and $R$ is its spine. (If such a path is trivial, which we allow, then its unique vertex lies on $R$ and also counts as a tooth; see Figure 8.1.1.)

## 数学代写|图论代写Graph Theory代写|Paths, trees, and ends

There are two fundamentally different aspects to the infinity of an infinite connected graph: one of ‘length’, expressed in the presence of rays, and one of ‘width’, expressed locally by infinite degrees. The infinity lemma tells us that at least one of these must occur:

Proposition 8.2.1. Every infinite connected graph has a vertex of infinite degree or contains a ray.

Proof. Let $G$ be an infinite connected graph with all degrees finite. Let $v_0$ be a vertex, and for every $n \in \mathbb{N}$ let $V_n$ be the set of vertices at distance $n$ from $v_0$. Induction on $n$ shows that the sets $V_n$ are finite, and hence that $V_{n+1} \neq \emptyset$ (because $G$ is infinite and connected). Furthermore, the neighbour of a vertex $v \in V_{n+1}$ on any shortest $v-v_0$ path lies in $V_n$. By Lemma 8.1.2, $G$ contains a ray.

Often it is useful to have more detailed information on how this ray or vertex of infinite degree lies in $G$. The following lemma enables us to find it ‘close to’ any given infinite set of vertices.
Lemma 8.2.2. (Star-Comb Lemma)
Let $U$ be an infinite set of vertices in a connected graph $G$. Then $G$ contains either a comb with all teeth in $U$ or a subdivision of an infinite star with all leaves in $U$.

Proof. As $G$ is connected, it contains a path between two vertices in $U$. This path is a tree $T \subseteq G$ every edge of which lies on a path in $T$ between two vertices in $U$. By Zorn’s lemma there is a maximal such tree $T^$. Since $U$ is infinite and $G$ is connected, $T^$ is infinite. If $T^*$ has a vertex of infinite degree, it contains the desired subdivided star.

Suppose now that $T^$ is locally finite. Then $T^$ contains a ray $R$ (Proposition 8.2.1). Let us construct a sequence $P_1, P_2, \ldots$ of disjoint $R-U$ paths in $T^$. Having chosen $P_i$ for every $i$ between two vertices in $U$; let us think of $P$ as traversing this edge in the same direction as $R$. Let $w$ be the last vertex of $v P$ on $v R$. Then $P_n:=w P$ contains an $R-U$ path, and $P_n \cap P_i=\emptyset$ for all $i<n$ because $P_i \cup R w \cup P_n$ contains no cycle.

## 数学代写|图论代写Graph Theory代写|Basic notions, facts and techniques

$$V=\left{x_0, x_1, x_2, \ldots\right} \quad E=\left{x_0 x_1, x_1 x_2, x_2 x_3, \ldots\right}$$

$$V=\left{\ldots, x_{-1}, x_0, x_1, \ldots\right} \quad E=\left{\ldots, x_{-1} x_0, x_0 x_1, x_1 x_2, \ldots\right}$$

## 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。