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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|图论作业代写Graph Theory代考|Embeddings in Robust Expanders with Medium Density

The robust sublinear expansion underpins all of our constructions of $H$-subdivisions when the graph $G$ is no longer dense. At a high level, in $G$, we anchor on some carefully chosen vertices and embed paths between anchors (corresponding to the edge set of $H$ ) one at a time. As these paths in the subdivision need to be internally vertex disjoint, to realise this greedy approach we will need to build a path avoiding a certain set of vertices. This set of vertices to avoid contains previous paths that we have already found and often some small set of ‘fragile’ vertices that we wish to keep free.

To carry out such robust connections, we use the small-diameter property of sublinear expanders. We aim to anchor at vertices with large ‘boundary’ compared to the total size of all paths needed, that is, being able to access many vertices within short distance. If there are $d$ vertices of sufficiently high degree, we can anchor on them. Assuming this is not the case essentially enables us to view $G$ as if it is a ‘relatively regular’ graph. We now use a web structure in which each core vertex is connected by a tree to a large ‘exterior’. Using the relative regularity of $G$, together with the fact that it is not too sparse, we can pull out many reasonably large stars and link them up to find webs. We then anchor on their core vertices and connect pairs via the exteriors of the corresponding webs, while being careful to avoid the fragile centre parts of other webs.

## 数学代写|图论作业代写Graph Theory代考|Embeddings in Sparse Robust Expanders

The method of building and connecting webs breaks down if the expander is too sparse, and we need to use other structures in this case.

For the easier problem of finding minors, it suffices to find $d$ large balls and link them up by internally disjoint paths according to the structure of $H$; contracting each ball gives $H$ as a minor. In order to be able to find the paths, we ensure the balls are sufficiently far apart that any given pair of balls can be expanded to very large size, avoiding all others, and then connect the pairs one by one.

Coming back to embedding $H$-subdivisions, we shall follow a similar general strategy. However, an immediate obstacle we encounter is that we need to be able to lead a constant number of paths arriving at each ball disjointly to the anchor vertex. In other words, each anchor vertex has to expand even after removing a constant number of disjoint paths starting from itself. Our expansion property is simply too weak for this.

We therefore use a new structure we call a ‘nakji’. Each nakji consists of several ‘legs’, which are balls pairwise far apart, linked to a central well-connected ‘head’. This structure is designed precisely to circumvent the above problem by doing everything in reverse order. Basically, instead of looking for anchor vertices that expand robustly, we rather anchor on nakjis and link them via their legs first and then extend the paths from the legs in each nakji’s head using connectivity. The remaining task is then to find many nakjis. This is done essentially by linking small subexpanders within $G$, after removing the few high-degree vertices.

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## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师