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Understanding Graph Data Structures

Handout
Grade 12
English

Understanding Graph Data Structures

Introduction to Graph Data Structures

A graph is a non-linear data structure consisting of a finite set of vertices (or nodes) and a set of edges that connect these vertices. Graphs are widely used to model real-world relationships and networks, such as social networks, transportation routes, and computer networks. Unlike linear data structures like arrays or linked lists, or tree structures, graphs allow for complex, many-to-many relationships between elements.

Key Components

  1. Vertices (Nodes): These are the fundamental entities in a graph. Each vertex can represent an object, a person, a city, or any other item in a dataset.
  1. Edges: These are the connections between vertices. An edge signifies a relationship or interaction between two vertices. Edges can have different properties, which leads to various types of graphs.

Types of Graphs

  • Undirected Graph: Edges have no direction. If an edge connects vertex A to vertex B, it implies that B is also connected to A. Think of a two-way street.
  • Directed Graph (Digraph): Edges have a specific direction. An edge from A to B does not necessarily mean there's an edge from B to A. This is like a one-way street.
  • Weighted Graph: Each edge is assigned a numerical value, or 'weight', representing a cost, distance, or strength of the connection. For example, in a road network, weights could be distances between cities.

Graph Representations

To store and manipulate graphs in computer memory, two primary methods are used:
  1. Adjacency Matrix: This is a square matrix where the rows and columns represent vertices. An entry
    
    is $ 1 $ (or the weight, for a weighted graph) if there's an edge from vertex $ i $ to vertex
    
    otherwise. This representation is suitable for dense graphs (many edges).
  1. Adjacency List: This is an array of lists. The index of the array represents a vertex, and each list at that index contains all the vertices directly connected to it. This is generally more space-efficient for sparse graphs (few edges).
Graphs are powerful tools for solving complex problems, from finding the shortest path between two points to analyzing data flow and dependencies.

Curriculum Expectations

  • Custom Standard 2