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

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## 数学代写|图论作业代写Graph Theory代考|Graphs

A graph $G$ is a pair of sets $(V, E)$, where $V$ is a finite non-empty set of elements called vertices, and $E$ is a finite set of elements called edges, each of which has two associated vertices. The sets $V$ and $E$ are the vertex-set and edge-set of $G$, and are sometimes denoted by $V(G)$ and $E(G)$. The number of vertices in $G$ is called the order of $G$ and is usually denoted by $n$ (but sometimes by $|G|$ or $|V(G)|$ ); the number of edges is denoted by $m$. A graph with only one vertex and no edges is called trivial.

An edge whose vertices coincide is a loop, and if two edges have the same pair of associated vertices, they are called multiple edges. In this book, unless otherwise specified, graphs are assumed to have no loops or multiple edges; that is, they are taken to be simple. Hence, an edge $e$ can be considered as its associated pair of vertices, $e={v, w}$, usually shortened to $v w$. An example of a graph of order 5 is shown in Fig. 1(a).

## 数学代写|图论作业代写Graph Theory代考|Adjacency and degrees

The vertices of an edge are its endpoints or ends, and the edge is said to join these vertices. An endpoint of an edge and the edge are incident with each other. Two vertices that are joined by an edge are called neighbours and are said to be adjacent; if $v$ and $w$ are adjacent vertices, we sometimes write $v \sim w$, and if they are not adjacent we write $v \nsim w$. Two edges are adjacent if they have a vertex in common.

The set $N(v)$ of neighbours of a vertex $v$ is called its neighbourhood. If $X \subset V$, then $N(X)$ denotes the set of vertices not in $X$ that are adjacent to some vertex of $X$. The closed neighbourhood of a vertex $v$ is defined as $N[v]=N(v) \cup{v}$. Two vertices $v$ and $w$ are true twins if $N[v]=N[w]$ and are false twins if $N(v)=N(w)$.

The degree deg $v$, or $d(v)$, of a vertex $v$ is the number of its neighbours; in a nonsimple graph, it is the number of occurrences of the vertex as an endpoint of an edge, with loops counted twice. A vertex of degree 0 is an isolated vertex and one of degree 1 is a pendant vertex. A graph is regular if all of its vertices have the same degree, and is $k$-regular if that degree is $k$; a 3-regular graph is sometimes called cubic. The maximum degree in a graph G is denoted by $\Delta(G)$ or just $\Delta$, and the minimum degree by $\delta(G)$ or $\delta$. The degree sequence of a graph is the non-increasing sequence of its vertex degrees, for example, [3,2,2,2,1] in both Fig. 1(a) and Fig. 1(b), although they are not the same graph. Determining whether a given sequence of numbers is the degree sequence of a simple graph can be done using an algorithm by Havel and Hakimi or a characterization theorem of Erdôs and Gallai.

## 数学代写|图论作业代写GRAPH THEORY代考|ADJACENCY AND DEGREES

An isomorphism between two graphs G and H is a bijection between their vertex-sets that preserves both adjacency and non-adjacency. The graphs G and H are isomorphic, written $G \cong H$, if there exists an isomorphism between them.

An automorphism of a graph $G$ is an isomorphism of $G$ with itself. The set of all automorphisms of a graph $G$ forms a group, called the automorphism group of $G$ and denoted by $\operatorname{Aut}(G)$.

A homomorphism of a graph $G$ to a graph $H$ is a mapping of the vertex-set of $G$ to the vertex-set of $H$ that preserves adjacency (but not necessarily non-adjacency). The graph $G$ is homomorphic to $H$ if there exists such a homomorphism.

## 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。