# 数学代写|图论作业代写Graph Theory代考|Structured families of graphs

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

A planar graph is one that can be embedded in the plane in such a way that no two edges meet except at a vertex that is incident with both. If a graph $G$ is embedded in this way, then the points of the plane not on $G$ are partitioned into open sets called faces or regions. Euler discovered the basic relationship between the numbers of vertices, edges and faces.

Euler’s formula Let $G$ be a connected graph embedded in the plane with $n$ vertices, $m$ edges and $f$ faces. Then $n-m+f=2$.

It follows from this result that a simple planar graph with $n$ vertices $(n \geq 3)$ has at most 3(n-2$edges, and at most 2(n-2) edges if it is bipartite. From this it follows that the two graphs$K_{5}$and$K_{3,3}$are non-planar. Kuratowski proved that these two graphs are the only barriers to planarity. Kuratowski’s theorem The following statements are equivalent for a graph$G$: • G is planar; • G has no subgraph that is homeomorphic to$K_{5}$or$K_{3,3}$; • G has no subgraph that is contractible to$K_{5}$or$K_{3,3}$## 数学代写|图论作业代写Graph Theory代考|Line graphs and intersection graphs The line graph$L(G)$of a graph$G$has the edges of$G$as its vertices, with two of these vertices adjacent if the corresponding edges are adjacent in$G$. An example is given in Fig. 8 . A graph is a line graph if and only if its edges can be partitioned into complete subgraphs in such a way that no vertex is in more than two of these subgraphs. Line graphs are also characterized by the property of having none of nine particular graphs as a forbidden subgraph. The chromatic index of a graph$G$equals the chromatic number of its line graph – that is,$\chi^{\prime}(G)=\chi(L(G))$. Intersection graphs generalize the idea behind line graphs. Let$\mathcal{S}=\left{S_{1}, S_{2}, \ldots, S_{n}\right}$be a collection of sets. The intersection graph of$\mathcal{S}$is the graph$G$obtained by assigning a distinct vertex$v_{i}$of$G$for each set$S_{i}$in$\mathcal{S}$and joining two vertices by an edge precisely when their corresponding sets have a non-empty intersection – that is,$v_{i} v_{j} \in E(G)$if and only if$i \neq j$and$S_{i} \cap S_{j} \neq \varnothing$. Line graphs are a particular class of intersection graphs, where the edge-set of the original graph plays the role of$\mathcal{S}$. Other important classes of intersection graphs are interval graphs intervals on a line, string graphs simple curves in the plane, circle graphs chords of a circle, EPT graphs edge intersection graphs of paths in a tree, unit cube graphs axis-oriented unit cubes in 3-space and dozens of other structured families of graphs. One of the oldest results on intersection graphs is that every graph is the intersection graph of its stars – that is, each$S_{i}$is the set of edges in$E(G)$incident with vertex$v_{i}$. ## 图论代写 ## 数学代写|图论作业代写GRAPH THEORY代考|PLANARITY 平面图是一种可以嵌入平面中的图形，即除了与两个边相交的顶点外，没有两条边相交。如果一个图表G以这种方式嵌入，则平面的点不在G被划分为称为面或区域的开放集。欧拉发现了顶点数、边数和面数之间的基本关系。 欧拉公式让G是嵌入平面中的连通图n顶点，米边缘和F面孔。然后n−米+F=2. 从这个结果可以得出一个简单的平面图n顶点(n≥3)最多有 3 个n−2$和dG和s,一种nd一种吨米这s吨2(n−2如果它是二分的，则边缘。由此可知，两张图ķ5和ķ3,3是非平面的。Kuratowski 证明了这两个图是平面性的唯一障碍。

Kuratowski 定理 以下陈述对于图是等价的G :

• G是平面的；
• G 没有同胚的子图ķ5或者ķ3,3;
• G 没有可收缩的子图ķ5或者ķ3,3

## 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。