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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|图论作业代写Graph Theory代考|Tree Properties

As finding a minimum spanning tree is (in graph theoretic terms) quick and easy, we focus our attention on the properties of trees and what a spanning tree can tell us about its graph. As is common in mathematics, the things with the simplest definitions provide an abundance of material to study in depth. Trees in particular provide ample opportunities to strengthen our proof writing skills, specifically induction and contradiction methods. We begin with some results that arise through counting techniques. Recall that a vertex of degree 1 is called a leaf.
Theorem 3.4 Every tree with at least two vertices has a leaf.
Proof: Suppose for a contradiction that there exists a tree $T$ with at least two vertices that does not contain a leaf. Since $T$ must be connected, we know no vertex has degree 0 , and therefore every vertex of $T$ must have degree at least 2. But then by Theorem $2.5$ we know $T$ must have a cycle, which contradicts that $T$ is acyclic. Thus $T$ must contain a leaf.

Exercise $3.18$ expands on this theorem to show that every tree (with at least two vertices) in fact has at least two leaves.

Beyond simply showing trees have a specific property, the result above allows us to do a remarkably useful thing-prune a tree! Recall that $G-v$ denotes removing the vertex from $G$ along with all edges incident to $v$. The proof of the following lemma appears in Exercise 3.16.
Lemma 3.5 Given a tree $T$ with a leaf $v$, the graph $T-v$ is still a tree.
Removing a leaf from a tree will always remove exactly one vertex and one edge, creating a tree with a smaller size. This technique naturally lends itself to induction arguments.

## 数学代写|图论作业代写Graph Theory代考|Tree Enumeration

In Section $1.6$ we discussed how a degree sequence can help determine the structure of a graph, though it may not be unique. As it turns out, there is a similar result for trees, but one in which the tree can be uniquely determined from the sequence given. Instead of focusing on degrees, this sequence, called a Prüfer sequence, uses the location of leaves within a tree. These sequences were introduced in 1918 by the German mathematician Ernst Paul Heinz Prüfer as a means for proving Cayley’s Theorem (see Theorem 3.14)[70].

Definition $3.13$ Given a tree $T$ on $n>2$ vertices (labeled $1,2, \ldots, n$ ), the Prüfer sequence of $T$ is a sequence $\left(s_1, s_2, \ldots, s_{n-2}\right)$ of length $n-2$ defined as follows:

• Let $l_1$ be the leaf of $T$ with the smallest label.
• Define $T_1$ to be $T-l_1$.
• For each $i \geq 1$, define $T_{i+1}=T_i-l_{i+1}$, where $l_{i+1}$ is the leaf with the smallest label of $T_i$.
• Define $s_i$ to be the neighbor of $l_i$.
While this definition of the Prüfer sequence is quite detailed, it is not too difficult to work with in practice. The main idea is to prune the leaf of the smallest index, while keeping track of its unique neighbor. At the end of the pruning, two adjacent vertices will remain.
• Looking back at this example, we should see a few items stand out. First, the leaves of $T$ are $1,4,5,7$, and 8 , none of which appear in the Prüfer sequence. This is because we are always noting the neighbor of a leaf, not the leaf itself. You should also notice that a vertex appears in the sequence one less than its degree. Also, the vertex with label $n$ will always be adjacent to the vertex appearing as $s_{n-2}$, which occurs since the last graph created $\left(T_{n-2}\right)$ will consist of only two vertices and $n$ will never be the smallest leaf remaining. Using these ideas, we can actually work backwards to build the tree from a given Prüfer sequence.

# 图论代考

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## 数学代写|图论作业代写Graph Theory代考|Tree Enumeration

. Tree Enumeration . Tree Enumeration

• 设$l_1$为标签最小的$T$的叶。
• 定义$T_1$为$T-l_1$
• 对于每个$i \geq 1$，定义$T_{i+1}=T_i-l_{i+1}$，其中$l_{i+1}$是带有$T_i$最小标签的叶子
• 定义$s_i$为$l_i$的邻居
虽然Prüfer序列的这个定义非常详细，但在实践中使用它并不太难。主要思想是修剪最小索引的叶，同时跟踪它唯一的邻居。在修剪结束时，两个相邻顶点将保留下来。回过头来看这个例子，我们应该会看到一些突出的项目。首先，$T$的叶子是$1,4,5,7$和8，它们都没有出现在Prüfer序列中。这是因为我们总是注意叶子的邻居，而不是叶子本身。您还应该注意到，一个顶点出现在比它的度数小1的序列中。此外，带有标签$n$的顶点将始终与显示为$s_{n-2}$的顶点相邻，这是因为创建的最后一个图$\left(T_{n-2}\right)$将只包含两个顶点，而$n$永远不会是剩余的最小叶。使用这些思想，我们实际上可以从给定的Prüfer序列逆向构建树。

## 有限元方法代写

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## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师