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Graph Data Structure Handout

Handout
Grade 12
English

Graph Data Structure Handout

Introduction

A graph is a fundamental data structure in computer science used to model pairwise relations between objects. Graphs are widely used in fields such as networking, biology, social sciences, and computer algorithms.

Components of a Graph

A graph consists of two main components:
  • Vertices (Nodes): These represent the entities or objects.
  • Edges (Links): These connect pairs of vertices and represent the relationships between them.
Graphs can be represented as:
  • Undirected Graphs: Edges have no direction (e.g., friendship networks).
  • Directed Graphs (Digraphs): Edges have a direction, indicating a one-way relationship (e.g., Twitter follower relationships).

Types of Graphs

  • Weighted Graph: Each edge has a numerical value or weight (e.g., distance, cost).
  • Unweighted Graph: All edges are equal, with no associated weights.
  • Cyclic Graph: Contains at least one cycle (a path where the start and end nodes are the same).
  • Acyclic Graph: Contains no cycles (e.g., trees, DAGs).
  • Connected Graph: There is a path between every pair of vertices.
  • Disconnected Graph: Not all vertices are connected.

Graph Representations

  • Adjacency Matrix: A 2D array where each cell [i][j] indicates if there is an edge from vertex i to vertex j.
  • Adjacency List: Each vertex stores a list of adjacent vertices. This is more space-efficient for sparse graphs.

Applications of Graphs

  • Social networks: Modeling relationships between people.
  • Web page linking: The structure of the internet.
  • Transportation networks: Roads, flight routes.
  • Dependency resolution: Task scheduling, course prerequisites.

Example: Adjacency List Representation

Suppose we have a graph with vertices A, B, C, and edges: (A, B), (A, C), (B, C):
  • A: B, C
  • B: C
  • C: (none)

Key Terms

  • Vertex (Node)
  • Edge (Link)
  • Path
  • Degree
  • Cycle
  • Connectedness
Important: Understanding graph structures is critical for solving complex problems in computer science, such as finding the shortest path, network flows, and traversals.

Graph Data Structure Handout

Introduction

A graph is a fundamental data structure in computer science used to model pairwise relations between objects. Graphs are widely used in fields such as networking, biology, social sciences, and computer algorithms.

Components of a Graph

A graph consists of two main components:
  • Vertices (Nodes): These represent the entities or objects.
  • Edges (Links): These connect pairs of vertices and represent the relationships between them.
Graphs can be represented as:
  • Undirected Graphs: Edges have no direction (e.g., friendship networks).
  • Directed Graphs (Digraphs): Edges have a direction, indicating a one-way relationship (e.g., Twitter follower relationships).

Types of Graphs

  • Weighted Graph: Each edge has a numerical value or weight (e.g., distance, cost).
  • Unweighted Graph: All edges are equal, with no associated weights.
  • Cyclic Graph: Contains at least one cycle (a path where the start and end nodes are the same).
  • Acyclic Graph: Contains no cycles (e.g., trees, DAGs).
  • Connected Graph: There is a path between every pair of vertices.
  • Disconnected Graph: Not all vertices are connected.

Graph Representations

  • Adjacency Matrix: A 2D array where each cell [i][j] indicates if there is an edge from vertex i to vertex j.
  • Adjacency List: Each vertex stores a list of adjacent vertices. This is more space-efficient for sparse graphs.

Applications of Graphs

  • Social networks: Modeling relationships between people.
  • Web page linking: The structure of the internet.
  • Transportation networks: Roads, flight routes.
  • Dependency resolution: Task scheduling, course prerequisites.

Example: Adjacency List Representation

Suppose we have a graph with vertices A, B, C, and edges: (A, B), (A, C), (B, C):
  • A: B, C
  • B: C
  • C: (none)

Key Terms

  • Vertex (Node)
  • Edge (Link)
  • Path
  • Degree
  • Cycle
  • Connectedness
Important: Understanding graph structures is critical for solving complex problems in computer science, such as finding the shortest path, network flows, and traversals.

Discussion Questions & Critical Thinking Prompts

  1. How might the choice between an adjacency matrix and an adjacency list affect the performance of graph algorithms in large, sparse networks? Explain your reasoning.
  1. Consider a transportation network (like city roads or airline routes). What type of graph (directed/undirected, weighted/unweighted, cyclic/acyclic) would best represent it, and why?
  1. In what ways could understanding cycles in a graph help in solving real-world problems, such as detecting fraud in financial transactions or scheduling tasks?
  1. Imagine you are designing a social media platform. How would you use graphs to model user interactions, and what challenges might arise when scaling to millions of users?
  1. Discuss how the concept of connectedness in graphs could be applied to analyzing the resilience of power grids or communication networks during failures.