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# 数学代写|图论代写graph theory代考|Kuratowski’s Theorem

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## 数学代写|图论代写graph theory代考|Euler’s Formula

One major result regarding planarity that is quite useful in gaining some intuition as to the planarity of a graph was proven in 1752 by a mathematician we spent an entire section discussing, Leonhard Euler. The result was given in more geometric terms (and planarity is one area of intersection between graph theory and geometry) and uses an additional term relating to the drawing of a graph, namely a region.

Given a planar drawing of a graph $G$, a region is a portion of the plane completely bounded by the edges of the graph.

In practice, we can usually see the regions of a graph fairly easily, as long as we do not forget the infinite (or exterior) region. For example, the following two graphs $G_{1}$ and $G_{2}$, each have 6 vertices, but $G_{1}$ has 9 edges and 5 regions whereas $G_{2}$ has 5 edges and only one region, the infinite one.

Note that every tree has exactly 1 region since no cycles exist to fully encompass a portion of the plane. As both of the graphs above are planar, they satisfy Euler’s Formula below.

## 数学代写|图论代写graph theory代考|Cycle-Chord Method

When a graph is drawn so the vertices are roughly arranged around a circle, it can often be easier to think about shifting their positions on the page or stretching the edges to obtain a planar drawing. But when the graph is drawn to highlight some other attribute, such as it being bipartite or showing some clumping of vertices, it can be challenging to find a planar drawing. The next few pages will detail one method for finding a planar drawing, called the CycleChord Method. The graph $G_{6}$ below will serve as an example of how to use this method.

To begin, put the vertices in a circular pattern, but with some care in their arrangement. We want to find a spanning cycle (also called a hamiltonian cycle) or something approximating a spanning cycle, when placing the vertices. The edges in bold on the left represent those that are currently being placed in the planar drawing; the gray edges are ones not yet placed.

## 数学代写|图论代写GRAPH THEORY代考|Proof of Kuratowski’s Theorem

Now that we have some familiarity with properties of planar graphs, we return to the proof of Kuratowski’s Theorem, which basically states that being nonplanar is equivalent to having one of two forbidden structures: subdivisions of $K_{5}$ or $K_{3,3}$. The formal statement of the theorem is below and is written as a biconditional. Recall that biconditional statements are special in that they show both necessary and sufficient conditions for property to hold. In this case, we need only to know if a graph contains a subdivision of $K_{3,3}$, or $K_{5}$ to determine its planarity.

## 数学代写|图论代写GRAPH THEORY代考|EULER’S FORMULA

1752 年，一位我们花了整整一节讨论的数学家莱昂哈德·欧拉（Leonhard Euler）证明了一个关于平面性的主要结果，它对于获得一些关于图形平面性的直觉非常有用。结果以更多的几何术语给出一种ndpl一种n一种r一世吨是一世s这n和一种r和一种这F一世n吨和rs和C吨一世这nb和吨在和和nGr一种pH吨H和这r是一种ndG和这米和吨r是并使用与绘制图形有关的附加术语，即区域。

## 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。