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# 数学代写|图论代写Graph Theory代考|Euler tours

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## 数学代写|图论代写Graph Theory代考|Constructing an Euler tour

Returning to the seven bridges of Königsberg, we can model the problem by representing each area separated by a bridge as a vertex, and each bridge by an edge connecting two separated areas, leading to the graph (with multiple edges) shown in Figure 4.2. The people of Königsberg were interested in finding a specific tour:

Definition 4.1: A tour of a graph $G$ is a $(u, v)$-walk in which $u=v$ (i.e., it is a closed walk) and that traverses each edge in G. An Euler tour is a tour in which all edges are traversed exactly once.

Euler tours were named after the Swiss mathematician Leonhard Euler who initially solved the problem of the Königsberg bridges. To this end, he proved the following theorem:

Theorem 4.1: A connected graph $G$ (with more than one vertex) has an Euler tour if and only if it has no vertices of odd degree.

## 数学代写|图论代写Graph Theory代考|The Chinese postman problem

Let us now consider a practical application of Euler’s research: the Chinese postman problem, so-called because it was first postulated by the Chinese mathematician Kuan [1962]. This problem is more general and also more complicated than that of finding an Euler tour. Consider a weighted graph $G$ in which each edge has a nonnegative weight. The problem is to find a closed walk $W=\left[v_{0}, e_{1}, v_{1} \ldots v_{n-1}, e_{n}, v_{n}\right]$ that covers all edges of $G$, but with minimal weight. In other words, $E(W)=E(G)$ and $\sum_{i=1}^{n} w\left(e_{i}\right)$ is minimal. Note that we do not demand that each edge is traversed exactly once, for in that case we would have an Euler tour, and obviously, such a walk would automatically have minimal weight. Instead, we are aiming for a closed walk such that if it is necessary to cross an edge more than once, that the walk is such that the total weight is kept as low as possible.

The Chinese postman problem is a generalization of many traversal problems. Consider the following examples.

## 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。