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

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## 数学代写|图论作业代写Graph Theory代考|Vertex-connectivity

A vertex v in a graph G is a cut-vertex if G-v has more components than G. For a connected graph, this is equivalent to saying that $G-v$ is disconnected, and that there exist vertices u and w, different from v, for which v is on every u-w path.

A non-trivial graph is non-separable if it is connected and has no cut-vertices. Note that under this definition the graph $K_{2}$ is non-separable. There are many characterizations of the other non-separable graphs, as the following statements are all equivalent for a connected graph $G$ with at least three vertices:

• G is non-separable;
• every two vertices of $G$ are on a cycle;
• every vertex and edge of $G$ are on a cycle;
• every two edges of $G$ are on a cycle;
• for any three vertices $u, v$ and $w$ in $G$, there is a v-w path that contains u;
• for any three vertices $u, v$ and $w$ in $G$, there is a v-w path that does not contain u;
• for any two vertices $v$ and $w$ and any edge $e$ in G, there is a v-w path that contains e.

## 数学代写|图论作业代写Graph Theory代考|Edge-connectivity

There is an analogous body of material that involves edges rather than vertices, and because of the similarities, we treat it in less detail.

An edge e is a cut-edge (or bridge) of a graph $G$ if $G-e$ has more components than G. In contrast to the situation with vertices, the removal of an edge cannot increase the number of components by more than 1. An edge $e$ is a cut-edge if and only if there exist vertices $v$ and $w$ for which $e$ is on every $v-w$ path. The cut-edges in a graph are also characterized by the property of not lying on a cycle; thus, a graph is a forest if and only if every edge is a cut-edge. Graphs having no cut-edges can be characterized in a variety of ways similar to those having no cut-vertices – that is, non-separable graphs. The concepts corresponding to cycles and paths for vertices are circuits and trails for edges.

Moving beyond cut-edges, we have the following definitions. A graph G is l-edgeconnected if the removal of fewer than $l$ edges always leaves a connected graph. Here is a third version of Menger’s theorem.

## 数学代写|图论作业代写GRAPH THEORY代考|VERTEX-CONNECTIVITY

• G 是不可分离的；
• 每两个顶点G处于一个循环中；
• 每个顶点和边G处于一个循环中；
• 每两个边缘G处于一个循环中；
• 对于任意三个顶点在,在和在在G, 有一条包含 u 的 vw 路径；
• 对于任意三个顶点在,在和在在G，有一条不包含u的vw路径；
• 对于任意两个顶点在和在和任何边缘和在 G 中，有一条包含 e 的 vw 路径。

## 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。
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