# 数学代写|图论作业代写Graph Theory代考|Flows in networks

#### Doug I. Jones

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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

(F1) $f(e, x, y)=-f(e, y, x)$表示所有$(e, x, y) \in \vec{E}$和$x \neq y$;

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