KS3 Maths: Graph Theory – Key Points | KS3 数学：图论考点精讲

📚 KS3 Maths: Graph Theory – Key Points | KS3 数学：图论考点精讲

Welcome to the world of graph theory! At KS3, you will encounter the basics of this fascinating branch of mathematics, which deals with networks of points and lines. Graphs help us solve real-world puzzles, from planning routes to understanding connections in social media. This revision guide covers the essential concepts, definitions, and typical problems you need to know.

欢迎来到图论的世界！在 KS3 阶段，你会接触到这个迷人数学分支的基础知识，它研究点和线构成的网络。图可以帮助我们解决现实世界中的谜题，从规划路线到理解社交媒体中的连接。本复习指南涵盖你需要掌握的核心概念、定义和典型问题。

1. What is a Graph? | 什么是图？

Graph theory is not about bar charts or line graphs – it is about a mathematical object called a graph. A graph consists of vertices (also called nodes or points) and edges (lines) that connect pairs of vertices.

图论并不是关于条形图或折线图——它是一种称为图的数学对象。图由顶点（也称为节点或点）和连接顶点对的边（线）组成。

Graphs are abstract representations of connections. For example, the vertices could represent towns, and the edges represent roads between them. This simple structure can model many situations: computer networks, family trees, and even the internet.

图是连接关系的抽象表示。例如，顶点可以代表城镇，边代表它们之间的道路。这种简单的结构可以模拟很多情况：计算机网络、家族树，甚至互联网。

In KS3 mathematics, we usually deal with simple graphs. A simple graph has no multiple edges between the same pair of vertices and no loops (an edge from a vertex back to itself).

在 KS3 数学中，我们通常处理简单图。简单图中没有连接同一对顶点的多重边，也没有环（从一个顶点出发回到自身的边）。

2. Vertices and Edges | 顶点与边

Each vertex is typically labelled with a capital letter or a number for easy reference. An edge can be described by naming its two endpoints, for instance, edge AB or edge (A, B). In an undirected graph, the order does not matter, so AB is the same as BA.

为了方便，每个顶点通常用大写字母或数字标记。一条边可以通过它的两个端点来描述，例如边 AB 或边 (A, B)。在无向图中，顺序无关紧要，所以 AB 和 BA 是一样的。

The total number of vertices in a graph is often denoted by V, and the number of edges by E. For a simple graph, the maximum number of edges is given by V(V−1)/2, which occurs when every vertex is connected to every other vertex – this is called a complete graph.

图中的顶点总数通常用 V 表示，边的数量用 E 表示。对于简单图，边的最大数量是 V(V−1)/2，这发生在每个顶点都与其他每个顶点相连时——这称为完全图。

3. Directed vs. Undirected Graphs | 有向图与无向图

In an undirected graph, edges have no direction; you can travel along them in either direction. Most basic graphs you encounter at KS3 are undirected.

在无向图中，边没有方向；你可以在任一方向沿着边走动。你在 KS3 遇到的大多数基础图都是无向图。

A directed graph, or digraph, has edges with arrows indicating a one-way connection. For example, a Twitter follow relationship is directed: A follows B does not mean B follows A.

有向图（简称有向图）的边带箭头，表示单向连接。例如，Twitter 的关注关系是有向的：A 关注 B 并不意味着 B 关注 A。

When you are asked about paths and degrees, you must first check whether the graph is undirected or directed, because the rules are slightly different.

当你被问到路径和度时，必须首先检查图是无向图还是有向图，因为规则略有不同。

4. Degree of a Vertex | 顶点的度

The degree of a vertex is the number of edges that are incident to it. In simple undirected graphs, it is simply how many edges touch that vertex. For example, a vertex that is the endpoint of three edges has degree 3.

顶点的度是指与该顶点相关联的边的数量。在简单的无向图中，它就是有几条边接触到这个顶点。例如，一个顶点作为三条边的端点，其度数为 3。

In a directed graph, we distinguish between in-degree (arrows

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