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# 数学代写|图论代考GRAPH THEORY代写|MATH2069 Applications

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## 数学代写|图论代写GRAPH THEORY代考|Applications

The Königsberg Bridge Problem can be viewed as the first application of graph theory, and in particular eulerian circuits. The remainder of this section will be devoted to a variety of problems whose solution can be found using an eulerian circuit. The examples chosen are by no means an exhaustive list, but rather a sampling of the wide-ranging applications of graph theory.
Snowplow Routes
Having grown up in the northeast, when a snowplow came down our road could have a major effect on morning commutes. If a city wants to optimize their snowplows, they would want to plan the routes so no plow would need to travel over a road more than once (assuming it takes only one pass to clear the street). This is a little different than finding a singular eulerian circuit in the graph, since we do not need one snowplow to cover all the streets, but rather we want to split a city into sectors that each snowplow would then travel using an eulerian circuit or trail.

For simplicity, assume the town of Crystal Spring has already determined the sectors for their plows; the following map shows one such sector. To turn this into a problem of finding an eulerian circuit, we first model the town as a graph where the edges represent street blocks and the vertices the street intersections.

## 数学代写|图论代写GRAPH THEORY代考|Chinese Postman Problem

In finding the eulerization of the Crystal Spring graph $G_1$, we didn’t distinguish between which edges to duplicate other than to minimize the overall total. In a town with a very regular grid structure traveling down one block versus another is inconsequential (think of Manhattan or Phoenix). However, for cities with more of an evolutionary development (such as Boston or Providence) or in rural towns where roads curve and blocks have different lengths, traveling down a stretch of road twice could look remarkably different from one choice to the next. How then would you model these differences? We add weights to each edge based on a chosen metric, such as distance, time or cost.
The weighted version of an eulerization problem is called the Chinese Postman Problem. The name originates not from anything particular about postmen in China, but rather from the mathematician who first proposed the problem – the Chinese mathematician Guan Meigu[42]. This problem first appeared in 1960, more than two centuries after Euler’s original paper! The full solution was published about a decade later, where the main idea is that a Postman delivering mail in a rural neighborhood should repeat the shortest stretches of road (provided any duplications are necessary). We will discuss the process for a small example, since we can usually find the best duplications by inspection. A more complete solution will be given in Section 5.2.2.

## 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。