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数学代写|离散数学作业代写discrete mathematics代考|Graph Theory
Graph theory is a practical branch of mathematics that deals with the arrangements of certain objects known as vertices (or nodes) and the relationships between them. It has been applied to practical problems such as the modelling of computer networks, determining the shortest driving route between two cities, the link structure of a website, the travelling salesman problem and the four-colour problem. ${ }^1$
Consider a map of the London underground, which is issued to users of the underground transport system in London. Then this map does not represent every feature of the city of London, as it includes only material that is relevant to the users of the London underground transport system. In this map the exact geographical location of the stations is unimportant, and the essential information is how the stations are interconnected to one another, as this allows a passenger to plan a route from one station to another. That is, the map of the London underground is essentially a model of the transport system that shows how the stations are interconnected.
The seven bridges of Königsberg ${ }^2$ (Fig. 9.1) is one of the earliest problems in graph theory. The city was set on both sides of the Pregel River in the early eighteenth century, and it consisted of two large islands that were connected to each other and the mainland by seven bridges. The problem was to find a walk through the city that would cross each bridge once and once only.
Euler showed that the problem had no solution, and his analysis helped to lay the foundations for graph theory as a discipline. This problem in graph theory is concerned with the question as to whether it is possible to travel along the edges of a graph starting from a vertex and returning to it and travelling along each edge exactly once. An Euler Path in a graph $\mathrm{G}$ is a simple path containing every edge of $\mathrm{G}$.
Euler noted, in effect, that for a walk through a graph traversing each edge exactly once depends on the degree of the nodes (i.e. the number of edges touching it). He showed that a necessary and sufficient condition for the walk is that the graph is connected and has zero or two nodes of odd degree. For the Köningberg graph, the four nodes (i.e. the land masses) have odd degree (Fig. 9.2).
A graph is a collection of objects that are interconnected in some way. The objects are typically represented by vertices (or nodes), and the interconnections between them are represented by edges (or lines). We distinguish between directed and adirected graphs, where a directed graph is mathematically equivalent to a binary relation, and an adirected (undirected) graph is equivalent to a symmetric binary relations.
数学代写|离散数学作业代写discrete mathematics代考|Undirected Graphs
An undirected graph (adirected graph) (Fig. 9.3) G is a pair of finite sets (V, E) such that $\mathrm{E}$ is a binary symmetric relation on $\mathrm{V}$. The set of vertices (or nodes) is denoted by $\mathrm{V}(\mathrm{G})$, and the set of edges is denoted by $\mathrm{E}(\mathrm{G})$.
A directed graph (Fig. 9.4) is a pair of finite sets (V, E) where E is a binary relation (that may not be symmetric) on V. A directed acylic graph (dag) is a directed graph that has no cycles. The example below is of a directed graph with three edges and four vertices.
An edge $e \in \mathrm{E}$ consists of a pair $\langle x, y>$ where $x, y$ are adjacent nodes in the graph. The degree of $x$ is the number of nodes that are adjacent to $x$. The set of edges is denoted by $E(G)$, and the set of vertices is denoted by $V(G)$.
A weighted graph is a graph $\mathrm{G}=(\mathrm{V}, \mathrm{E})$ together with a weighting function $w: \mathrm{E} \rightarrow \mathbb{N}$, which associates a weight with every edge in the graph. A weighting function may be employed in modelling computer networks: for example, the weight of an edge may be applied to model the bandwidth of a telecommunications link between two nodes. Another application of the weighting function is in determining the distance (or shortest path) between two nodes in the graph (where such a path exists).
For an directed graph, the weight of the edge is the same in both directions: i.e. $w\left(v_i, v_j\right)=w\left(v_j, v_i\right)$ for all edges $\left\langle v_i, v_j>\right.$ in the graph $\mathrm{G}$, whereas the weights may be different for a directed graph.
Two vertices $x, y$ are adjacent if $x y \in \mathrm{E}$, and $x$ and $y$ are said to be incident to the edge $x y$. A matrix may be employed to represent the adjacency relationship.
离散数学代写
数学代写|离散数学作业代写离散数学代考|图论
图论是数学的一个实用分支，它研究被称为顶点(或节点)的某些对象的排列以及它们之间的关系。它已被应用于一些实际问题，如计算机网络的建模、确定两个城市之间最短的驾驶路线、网站的链接结构、旅行推销员问题和四色问题。${ }^1$
考虑一张伦敦地铁的地图，它是发给伦敦地铁系统的用户的。因此，这张地图并不代表伦敦城市的每一个特征，因为它只包括与伦敦地下交通系统的用户相关的材料。在这张地图上，车站的确切地理位置并不重要，重要的信息是车站之间是如何相互连接的，因为这可以让乘客规划从一个车站到另一个车站的路线。也就是说，伦敦地铁地图本质上是一个交通系统的模型，展示了车站是如何相互连接的 Königsberg ${ }^2$的七座桥(图9.1)是图论中最早的问题之一。18世纪早期，这座城市坐落在普雷盖尔河的两岸，由两座大岛组成，它们通过七座桥相互连接，并与大陆相连。问题是要找到一条穿过城市的步道，每座桥只能穿过一次 欧拉表明这个问题没有解，他的分析帮助奠定了图论作为一门学科的基础。图论中的这个问题是关于是否有可能从一个顶点开始沿着图的边缘移动，然后返回到它，并沿着每条边只移动一次。图$\mathrm{G}$中的欧拉路径是包含$\mathrm{G}$的每条边的简单路径。
欧拉注意到，实际上，在图中遍历每条边仅一次取决于节点的度数(即与之接触的边的数量)。他证明了行走的一个充分必要条件是图是连通的并且有0个或2个奇数次节点。对于Köningberg图，四个节点(即陆地块)具有奇数度(图9.2) 图是以某种方式相互连接的对象的集合。对象通常由顶点(或节点)表示，它们之间的相互连接由边(或线)表示。我们区分了有向图和有向图，其中有向图在数学上等价于二进制关系，而有向(无向)图等价于对称二进制关系
数学代写|离散数学作业代写离散数学代考|无向图
无向图(有向图)(图9.3)G是一对有限集(V, E)，使得$\mathrm{E}$是$\mathrm{V}$上的二元对称关系。顶点(或节点)的集合用$\mathrm{V}(\mathrm{G})$表示，边的集合用$\mathrm{E}(\mathrm{G})$表示 有向图(图9.4)是一对有限集(V, E)，其中E是V上的二元关系(可能不对称)。有向酰基图(dag)是没有环的有向图。下面的例子是一个有向图，有三条边和四个顶点 边$e \in \mathrm{E}$由一对$\langle x, y>$组成，其中$x, y$是图中相邻的节点。$x$的度数是与$x$相邻的节点数。边集用$E(G)$表示，顶点集用$V(G)$表示。
加权图是一个图$\mathrm{G}=(\mathrm{V}, \mathrm{E})$加上一个加权函数$w: \mathrm{E} \rightarrow \mathbb{N}$，该函数将一个权值与图中的每条边关联起来。权重函数可用于计算机网络的建模:例如，边的权重可用于模拟两个节点之间的电信链路的带宽。权函数的另一个应用是确定图中两个节点之间的距离(或最短路径)(其中存在这样的路径)
对于有向图，边的权值在两个方向上都是相同的:例如，对于图$\mathrm{G}$中的所有边$\left\langle v_i, v_j>\right.$，边的权值都是$w\left(v_i, v_j\right)=w\left(v_j, v_i\right)$，而对于有向图，边的权值可能是不同的
如果$x y \in \mathrm{E}$，那么两个顶点$x, y$是相邻的，并且$x$和$y$被认为是$x y$边的事件。可以用一个矩阵来表示邻接关系
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