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数学代写|组合学代写Combinatorics代考|COMP418 Paths and Connected Graphs

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数学代写|组合学代写Combinatorics代考|Paths and Connected Graphs

Let $G$ be a graph with the set of vertices $V$, and the set of edges $E$. A walk in $\operatorname{graph} G$ is a finite sequence
$$W=v_{0} e_{1} v_{1} e_{2} \ldots v_{n-1} e_{n} v_{n},$$
where $v_{0}, v_{1}, \ldots, v_{n} \in V$, and $e_{1}, e_{2}, \ldots, e_{n} \in E$, and for any $i \in{1,2, \ldots, n}$, the vertices $v_{i-1}$ and $v_{i}$ are incident to the edge $e_{i}$. The vertices $v_{0}$ and $v_{k}$ are called, respectively, the origin, and the terminus of walk $W$, while the vertices $v_{1}, v_{2}, \ldots, v_{k-1}$ are called internal vertices of the walk.

The number of edges in the walk is called the length of the walk. Origin $v_{0}$ and terminus $v_{n}$ of walk $W=v_{0} e_{1} v_{1} e_{2} \ldots v_{n-1} e_{n} v_{n}$, need not be distinct. If $v_{0}=v_{n}$, then we say that walk $W$ is closed. If $v_{0} \neq v_{n}$, then walk $W$ is open. A vertex $v_{0}$ can be considered as a trivial walk.

A trail is a walk $W=v_{0} e_{1} v_{1} e_{2} \ldots v_{n-1} e_{n} v_{n}$ without repetition in the sequence of its edges $e_{1}, e_{2}, \ldots, e_{n}$.

If the vertices $v_{0}, v_{1}, \ldots, v_{n}$ of walk $W=v_{0} e_{1} v_{1} e_{2} \ldots v_{n-1} e_{n} v_{n}$ are distinct, then $W$ is called a path from $v_{0}$ to $v_{n}$.

数学代写|组合学代写Combinatorics代考|Isomorphic Graphs

Let $G$ be a graph with the set of vertices $V$, and the set of edges $E$, and let $G^{\prime}$ be a graph with the set of vertices $V^{\prime}$, and the set of edges $E^{\prime}$. The graphs $G$ and $G^{\prime}$ are isomorphic if there exist two bijective functions
$$f: V \rightarrow V^{\prime}, \quad g: E \rightarrow E^{\prime}$$

such that for each $v \in V$, and each $e \in E$, if $v$ is an endpoint of the edge $e$, then $f(v)$ is andpoint of the edge $g(e)$.

Example 8.5.1. Let $V={1,2,3,4,5}$ be the set of vertices, and ${i, j}$ be the edge with incident vertices $i$ and $j$, for any $i, j \in V$. Let us define $G_{1}$ and $G_{2}$ to be graphs with the same set of vertices $V$, and with the sets of edges $E_{1}=\left{e_{1}, e_{2}, e_{3}, e_{4}, e_{5}\right}$, and $E_{2}=\left{\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}, \varepsilon_{5}\right}$, respectively, where:
\begin{aligned} &e_{1}={1,3}, \quad e_{2}={2,4}, \quad e_{3}={3,5}, \quad e_{4}={4,1}, \quad e_{5}={5,2}, \ &\varepsilon_{1}={1,2}, \quad \varepsilon_{2}={2,3}, \quad \varepsilon_{3}={3,4}, \quad \varepsilon_{4}={4,5}, \quad \varepsilon_{5}={5,1} . \end{aligned}
Graphs $G_{1}$ nd $G_{2}$ are given in Figures $8.5 .1$ and 8.5.2, where every edge ${i, j}$ is presented as a segment with endpoints $i$ and $j$.

It is obvious that graphs $G_{1}$ and $G_{2}$ are complements to each other. We shall prove that these two graphs are also isomorphic.

数学代写|组合学代写Combinatorics代考|Paths and Connected Graphs

$$W=v_{0} e_{1} v_{1} e_{2} \ldots v_{n-1} e_{n} v_{n},$$

数学代写|组合学代写Combinatorics代考|lsomorphic Graphs

$$f: V \rightarrow V^{\prime}, \quad g: E \rightarrow E^{\prime}$$

$\backslash$ left 的分隔符缺失或无法识别，，分别，其中:
$e_{1}=1,3, \quad e_{2}=2,4, \quad e_{3}=3,5, \quad e_{4}=4,1, \quad e_{5}=5,2, \quad \varepsilon_{1}=1,2, \quad \varepsilon_{2}=2,3, \quad \varepsilon_{3}=3,4, \quad \varepsilon_{4}=4,5, \quad \varepsilon_{5}=5,1 .$

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。