# 数学代写|离散数学作业代写discrete mathematics代考|MPCS50103

## 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|无向图

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