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• 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|Hamiltonian Cycles

Think back to the city of Königsberg. The previous section determined when a graph would contain an eulerian circuit, a special type of circuit that must travel through every edge and vertex. This concept arose from a désire to cross every bridge in the city.
What if we change the requirements ever so slightly so that we are only concerned with the landmasses? This could model a delivery service with customers in every sector of the city. In graph theoretic terms, we are looking for a tour through the graph that hits every vertex exactly once. An example of such a tour on the graph representing Königsberg is shown above. What type of tour is this? If we need to start and end at the same location, we are searching for a cycle. If the starting and ending points can differ, we are searching for a path.
Definition 2.10 A cycle in a graph $G$ that contains every vertex of $G$ is called a hamiltonian cycle. A path that contains every vertex is called a hamiltonian path. A graph that contains a hamiltonian cycle is called hamiltonian.

Recall that a cycle or a path can only pass through a vertex once, so the hamiltonian cycles and paths travel through every vertex exactly once. Moreover, using the language of Definition 1.5, we could describe hamiltonian cycles and paths as spanning cycles and paths since they must include all vertices of the graph.

As with eulerian circuits, these specific cycles (or paths) are named for the mathematician who first formalized them, Sir William Hamilton. Hamilton posed this idea in 1856 in terms of a puzzle, which he later sold to a game dealer. The “Icosian Game” was a wooden puzzle with numbered ivory pegs where the player was tasked with inserting the pegs so that following them in order would traverse the entire board (shown on the following page). Perhaps not too surprisingly, this gamé was not a big money maker.

It should be noted that T.P. Kirkman, a contemporary of Hamilton’s, did much of the early work in the study of hamiltonian circuits. Whereas Hamilton primarily focused on one graph, Kirkman was concerned with the conditions that will guarantee a graph has a hamiltonian cycle. However, Hamilton deserves credit for publicizing the concept of a cycle that hits every vertex exactly once. This section will explore when a graph has a hamiltonian cycle and how to find an optimal, or near optimal, hamiltonian cycle.

## 数学代写|图论作业代写Graph Theory代考|The Traveling Salesman Problem

The discussion above should make clear the difficulty in determining if a graph is hamiltonian. But what if a graph is know to have a hamiltonian cycle? For example, every complete graph $K_n$ (for $n \geq 3$ ) must contain a hamiltonian cycle since it satisfies the criteria of Dirac’s Theorem. In this scenario, finding a hamiltonian cycle is quite elementary, and so, as mathematicians do, we generalize the problem to one in which the edges are no longer equivalent and have a weight associated to them. Then instead of asking whether a graph simply has a hamiltonian cycle, we can now ask how do we find the best hamiltonian cycle.

Historically, the extensive study of hamiltonian circuits arose in part from a simple question: A traveling salesman has customers in numerous cities; he must visit each of them and return home, but wishes to do this with the least total cost; determine the cheapest route possible for the salesman. In fact, Proctor and Gamble can be credited with the modern study of hamiltonian circuits when they sponsored a seemingly innocent competition in the 1960s asking for a shortest hamiltonian circuit visiting 33 cities across the United States. Mathematicians were intrigued and an entire branch of mathematics and computer science developed. For over half a century, some of the brightest minds have tackled the Traveling Salesman Problem (my graph theory professor in college called it “the disease”) and numerous books and websites are devoted to finding an optimal solution to both the general question and to specific instances (such as a cycle through all cities in Sweden). A full discussion of the problem is beyond the scope of this book, though you are encouraged to peruse [16] or [17]. You could honestly spend a semester just discussing the various algorithms, so we restrict ourselves to just a handful of these, with plenty of examples and exercises.

# 图论代考

## 有限元方法代写

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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