数学代写|图论作业代写Graph Theory代考|Paths and cycles

Doug I. Jones

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数学代写|图论作业代写Graph Theory代考|Paths and cycles

A path is a non-empty graph $P=(V, E)$ of the form
$$V=\left{x_0, x_1, \ldots, x_k\right} \quad E=\left{x_0 x_1, x_1 x_2, \ldots, x_{k-1} x_k\right},$$
where the $x_i$ are all distinct. The vertices $x_0$ and $x_k$ are linked by $P$ and are called its ends; the vertices $x_1, \ldots, x_{k-1}$ are the inner vertices of $P$. The number of edges of a path is its length, and the path of length $k$ is denoted by $P^k$. Note that $k$ is allowed to be zero; thus, $P^0=K^1$.

We often refer to a path by the natural sequence of its vertices, ${ }^3$ writing, say, $P=x_0 x_1 \ldots x_k$ and calling $P$ a path from $x_0$ to $x_k$ (as well as between $x_0$ and $x_k$ ).

For $0 \leqslant i \leqslant j \leqslant k$ we write
$x P$
\begin{aligned} P x_i & :=x_0 \ldots x_i \ x_i P & :=x_i \ldots x_k \ x_i P x_j & :=x_i \ldots x_j \end{aligned}
and
\begin{aligned} \stackrel{\circ}{P} & :=x_1 \ldots x_{k-1} \ P \stackrel{\circ}{x}i & :=x_0 \ldots x{i-1} \ \stackrel{\circ}{i}i P & :=x{i+1} \ldots x_k \ \stackrel{\circ}{x}i P \stackrel{\circ}{j}_j & :=x{i+1} \ldots x_{j-1} \end{aligned}
for the appropriate subpaths of $P$. We use similar intuitive notation for the concatenation of paths; for example, if the union $P x \cup x Q y \cup y R$ of three paths is again a path, we may simply denote it by $P x Q y R$.

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

A non-empty graph $G$ is called connected if any two of its vertices are linked by a path in $G$. If $U \subseteq V(G)$ and $G[U]$ is connected, we also call $U$ itself connected (in $G$ ). Instead of ‘not connected’ we usually say ‘disconnected’.

Proposition 1.4.1. The vertices of a connected graph $G$ can always be enumerated, say as $v_1, \ldots, v_n$, so that $G_i:=G\left[v_1, \ldots, v_i\right]$ is connected for every $i$.

Proof. Pick any vertex as $v_1$, and assume inductively that $v_1, \ldots, v_i$ have been chosen for some $i<|G|$. Now pick a vertex $v \in G-G_i$. As $G$ is connected, it contains a $v-v_1$ path $P$. Choose as $v_{i+1}$ the last vertex of $P$ in $G-G_i$; then $v_{i+1}$ has a neighbour in $G_i$. The connectedness of every $G_i$ follows by induction on $i$.

Let $G=(V, E)$ be a graph. A maximal connected subgraph of $G$ is called a component of $G$. Note that a component, being connected, is always non-empty; the empty graph, therefore, has no components.

If $A, B \subseteq V$ and $X \subseteq V \cup E$ are such that every $A-B$ path in $G$ contains a vertex or an edge from $X$, we say that $X$ separates the sets $A$ and $B$ in $G$. Note that this implies $A \cap B \subseteq X$. More generally we say that $X$ separates $G$ if $G-X$ is disconnected, that is, if $X$ separates in $G$ some two vertices that are not in $X$. A separating set of vertices is a separator. Separating sets of edges have no generic name, but some such sets do; see Section 1.9 for the definition of cuts and bonds. A vertex which separates two other vertices of the same component is a cutvertex, and an edge separating its ends is a bridge. Thus, the bridges in a graph are precisely those edges that do not lie on any cycle.

图论代考

数学代写|图论作业代写Graph Theory代考|Paths and cycles

$$V=\left{x_0, x_1, \ldots, x_k\right} \quad E=\left{x_0 x_1, x_1 x_2, \ldots, x_{k-1} x_k\right},$$
$x_i$都是不同的。顶点$x_0$和$x_k$由$P$连接，称为其端点;顶点$x_1, \ldots, x_{k-1}$是$P$的内部顶点。一条路径的边数就是它的长度，长度为$k$的路径用$P^k$表示。请注意，$k$允许为零;因此，$P^0=K^1$。

$x P$
\begin{aligned} P x_i & :=x_0 \ldots x_i \ x_i P & :=x_i \ldots x_k \ x_i P x_j & :=x_i \ldots x_j \end{aligned}

\begin{aligned} \stackrel{\circ}{P} & :=x_1 \ldots x_{k-1} \ P \stackrel{\circ}{x}i & :=x_0 \ldots x{i-1} \ \stackrel{\circ}{i}i P & :=x{i+1} \ldots x_k \ \stackrel{\circ}{x}i P \stackrel{\circ}{j}j & :=x{i+1} \ldots x{j-1} \end{aligned}

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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。

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