如果你也在 怎样代写图论Graph Theory 这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。图论Graph Theory有趣的部分原因在于,图可以用来对某些问题中的情况进行建模。这些问题可以在图表的帮助下进行研究(并可能得到解决)。因此,图形模型在本书中经常出现。然而,图论是数学的一个领域,因此涉及数学思想的研究-概念和它们之间的联系。我们选择包含的主题和结果是因为我们认为它们有趣、重要和/或代表主题。
图论Graph Theory通过熟悉许多过去和现在对图论的发展负责的人,可以增强对图论的欣赏。因此,我们收录了一些关于“图论人士”的有趣评论。因为我们相信这些人是图论故事的一部分,所以我们在文中讨论了他们,而不仅仅是作为脚注。我们常常没有认识到数学是一门有生命的学科。图论是人类创造的,是一门仍在不断发展的学科。
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数学代写|图论作业代写Graph Theory代考|Flows in networks
In this section we give a brief introduction to the kind of network flow theory that is now a standard proof technique in areas such as matching and connectivity. By way of example, we shall prove a classic result of this theory, the so-called max-flow min-cut theorem of Ford and Fulkerson. This theorem alone implies Menger’s theorem without much difficulty (Exercise 3), which indicates some of the natural power lying in this approach.
Consider the task of modelling a network with one source $s$ and one sink $t$, in which the amount of flow through a given link between two nodes is subject to a certain capacity of that link. Our aim is to determine the maximum net amount of flow through the network from $s$ to $t$. Somehow, this will depend both on the structure of the network and on the various capacities of its connections – how exactly, is what we wish to find out.
Let $G=(V, E)$ be a multigraph, $s, t \in V$ two fixed vertices, and c: $\vec{E} \rightarrow \mathbb{N}$ a map; we call $c$ a capacity function on $G$, and the tuple $N:=(G, s, t, c)$ a network. Note that $c$ is defined independently for the two directions of an edge. A function $f: \vec{E} \rightarrow \mathbb{R}$ is a flow in $N$ if it satisfies the following three conditions (Fig. 6.2.1):
(F1) $f(e, x, y)=-f(e, y, x)$ for all $(e, x, y) \in \vec{E}$ with $x \neq y$;
$\left(\mathrm{F}^{\prime}\right) f(v, V)=0$ for all $v \in V \backslash{s, t}$;
(F3) $f(\vec{e}) \leqslant c(\vec{e})$ for all $\vec{e} \in \vec{E}$.
数学代写|图论作业代写Graph Theory代考|Group-valued flows
Let $G=(V, E)$ be a multigraph and $H$ an abelian group. If $f$ and $g$ are two $H$-circulations then, clearly, $(f+g): \vec{e} \mapsto f(\vec{e})+g(\vec{e})$ and $-f: \vec{e} \mapsto-f(\vec{e})$ are again $H$-circulations. The $H$-circulations on $G$ thus form a group in a natural way.
A function $f: \vec{E} \rightarrow H$ is nowhere zero if $f(\vec{e}) \neq 0$ for all $\vec{e} \in \vec{E}$. An $H$-circulation that is nowhere zero is called an $H$-flow. ${ }^4$ Note that the set of $H$-flows on $G$ is not closed under addition: if two $H$-flows add up to zero on some edge $\vec{e}$, then their sum is no longer an $H$-flow. By Corollary 6.1.2, a graph with an $H$-flow cannot have a bridge.
For finite groups $H$, the number of $H$-flows on $G$-and, in particular, their existence surprisingly depends only on the order of $H$, not on $H$ itself:
Theorem 6.3.1. (Tutte 1954)
For every multigraph $G$ there exists a polynomial $P$ such that, for any finite abelian group $H$, the number of $H$-flows on $G$ is $P(|H|-1)$.
Proof. Let $G=:(V, E)$; we use induction on $m:=|E|$. Let us assume first that all the edges of $G$ are loops. Then, given any finite abelian group $H$, every map $\vec{E} \rightarrow H \backslash{0}$ is an $H$-flow on $G$. Since $|\vec{E}|=|E|$ when all edges are loops, there are $(|H|-1)^m$ such maps, and $P:=x^m$ is the polynomial sought.
Now assume there is an edge $e_0=x y \in E$ that is not a loop; let $\vec{e}_0:=\left(e_0, x, y\right)$ and $E^{\prime}:=E \backslash\left{e_0\right}$. We consider the multigraphs
$$
G_1:=G-e_0 \text { and } G_2:=G / e_0 .
$$
By the induction hypothesis, there are polynomials $P_i$ for $i=1,2$ such that, for any finite abelian group $H$ and $k:=|H|-1$, the number of $H$-flows on $G_i$ is $P_i(k)$. We shall prove that the number of $H$-flows on $G$ equals $P_2(k)-P_1(k)$; then $P:=P_2-P_1$ is the desired polynomial.
Let $H$ be given, and denote the set of all $H$-flows on $G$ by $F$. We are trying to show that
$$
|F|=P_2(k)-P_1(k) .
$$
图论代考
数学代写|图论作业代写Graph Theory代考|Flows in networks
在本节中,我们将简要介绍一种网络流理论,它现在是匹配和连通性等领域的标准证明技术。通过例子,我们将证明该理论的一个经典结果,即Ford和Fulkerson的所谓最大流量最小切定理。这个定理毫不费力地暗示了门格尔定理(练习3),这表明了这种方法的一些天然力量。
考虑一个具有一个源$s$和一个汇$t$的网络建模任务,其中通过两个节点之间的给定链接的流量取决于该链接的一定容量。我们的目标是确定从$s$到$t$通过网络的最大净流量。在某种程度上,这将取决于网络的结构及其连接的各种能力——具体如何,是我们希望找出的。
设$G=(V, E)$为多图,$s, t \in V$为两个固定顶点,c: $\vec{E} \rightarrow \mathbb{N}$为地图;我们将$c$称为$G$上的容量函数,将元组$N:=(G, s, t, c)$称为网络。请注意,对于一条边的两个方向,$c$是独立定义的。函数$f: \vec{E} \rightarrow \mathbb{R}$满足以下三个条件,即为$N$中的流(图6.2.1):
(F1) $f(e, x, y)=-f(e, y, x)$表示所有$(e, x, y) \in \vec{E}$和$x \neq y$;
所有人$\left(\mathrm{F}^{\prime}\right) f(v, V)=0$$v \in V \backslash{s, t}$;
(F3) $f(\vec{e}) \leqslant c(\vec{e})$为所有$\vec{e} \in \vec{E}$。
数学代写|图论作业代写Graph Theory代考|Group-valued flows
设$G=(V, E)$为多图,$H$为阿贝尔群。如果$f$和$g$是两个$H$ -循环,那么很明显,$(f+g): \vec{e} \mapsto f(\vec{e})+g(\vec{e})$和$-f: \vec{e} \mapsto-f(\vec{e})$又是$H$ -循环。因此,$G$上的$H$ -循环以一种自然的方式形成一个群体。
如果一个函数$f: \vec{E} \rightarrow H$对于所有的$\vec{e} \in \vec{E}$都是$f(\vec{e}) \neq 0$,那么它就不可能是零。任何地方都不是零的$H$循环称为$H$流。${ }^4$请注意,$G$上的$H$ -流的集合在加法下不是封闭的:如果两个$H$ -流在某些边$\vec{e}$上相加为零,那么它们的和不再是$H$ -流。根据推论6.1.2,具有$H$ -流的图不能有桥。
对于有限群$H$, $G$上的$H$流的数量,特别是,它们的存在令人惊讶地只取决于$H$的顺序,而不取决于$H$本身:
定理6.3.1。(Tutte, 1954)
对于每一个多图$G$,存在一个多项式$P$,使得对于任何有限阿贝尔群$H$, $G$上的$H$ -流的个数为$P(|H|-1)$。
证明。让$G=:(V, E)$;我们在$m:=|E|$上使用归纳。首先假设$G$的所有边都是循环。然后,给定任意有限阿贝尔群$H$,每个映射$\vec{E} \rightarrow H \backslash{0}$都是$G$上的$H$ -流。因为$|\vec{E}|=|E|$当所有的边都是环时,有$(|H|-1)^m$这样的映射,$P:=x^m$是要寻找的多项式。
现在假设有一条边$e_0=x y \in E$它不是一个循环;让$\vec{e}_0:=\left(e_0, x, y\right)$和$E^{\prime}:=E \backslash\left{e_0\right}$。我们考虑多重图
$$
G_1:=G-e_0 \text { and } G_2:=G / e_0 .
$$
根据归纳假设,对于$i=1,2$存在多项式$P_i$,使得对于任何有限阿贝尔群$H$和$k:=|H|-1$, $G_i$上的$H$ -flow的数量为$P_i(k)$。我们将证明$G$上的$H$流数等于$P_2(k)-P_1(k)$;那么$P:=P_2-P_1$就是我们想要的多项式。
设$H$给定,并用$F$表示$G$上所有$H$流的集合。我们正试图证明这一点
$$
|F|=P_2(k)-P_1(k) .
$$
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