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Adjacency matrix

A matrix with 1 for an edge between two vertices.

Adjacency matrix

An adjacency matrix stores the graph as a V×V matrix. If matrix[i][j] = 1 there is an edge from i to j; if 0 there is not.

Memory is O(V^2): suitable for dense graphs, wasteful for sparse ones. Neighbor query is O(1) and very fast.

In a weighted graph matrix[i][j] is the edge weight; 0 or infinity is used when there is no edge.

Comparison

The adjacency list uses less memory on sparse graphs (O(V+E)) and is efficient. The adjacency matrix suits dense graphs (O(V^2)).

With a matrix neighbor query is O(1); with a list O(degree). With a matrix scanning all neighbors is O(V); with a list O(degree), much faster.

The choice depends on the graph's density and the type of operation; it must be made carefully.

Practice Count neighbors

Write count_neighbors(graph, node). Return the node's neighbor count. Print 2 with g = {'a': ['b', 'c']}; print(count_neighbors(g, 'a')).

Starter code

solution.py
def count_neighbors(graph, node):
    # Count neighbors

Sample solution

solution.py
def count_neighbors(graph, node):
    # Count neighbors
    return len(graph[node])
g = {'a': ['b', 'c']}
print(count_neighbors(g, 'a'))

Expected output

2

Questions and answers

Review the lesson questions together with their answers. To test yourself, solve the lesson interactively.

1 Adjacency matrix: match the terms

Term Meaning
Adjacency matrix O(V^2)
Adjacency list O(V+E)
Dense graph Matrix
Sparse graph List

2 Best representation for a sparse graph?

  • Adjacency matrix
  • Hash table
  • Array
  • Adjacency list Correct answer

Explanation: Adjacency list

3 Matrix access

Question 
matrix[____][j] = 1
Answer 
matrix[i][j] = 1

Explanation: i is right.

All lessons in Intermediate

1 Merge sort

  1. Divide and conquer
  2. Merge operation
  3. Full merge sort
  4. Summary - Merge sort

2 Quick sort

  1. Pivot selection
  2. Partitioning
  3. Recursive quick sort
  4. Summary - Quick sort

3 Heap sort

  1. What is a heap?
  2. Heapify
  3. Build heap and sort
  4. Summary - Heap sort

4 Counting sort

  1. Counting idea
  2. Count array
  3. Output array
  4. Summary - Counting sort

5 Radix sort

  1. Radix idea
  2. Digit pass
  3. Stable sub-sort
  4. Summary - Radix sort

6 Sorting project

  1. Project: Overview
  2. Score array
  3. Apply merge sort
  4. Print statistics

7 Tree

  1. What is a tree?
  2. Node and edge
  3. Binary tree
  4. Summary - Trees

8 Binary search tree

  1. BST rule
  2. BST insert
  3. BST search
  4. Summary - BST

9 AVL tree

  1. Balanced tree
  2. Rotation
  3. Balance after insert
  4. Summary - AVL tree

10 Heap

  1. Min and max heap
  2. Insert
  3. Extract
  4. Summary - Heap

11 Priority queue

  1. What is a priority queue?
  2. Heap-based implementation
  3. Push and pop
  4. Summary - Priority queue

12 Data structures project

  1. Project: Overview
  2. Undo stack
  3. Pending queue
  4. Print the state

13 Graph basics

  1. What is a graph?
  2. Adjacency list
  3. Adjacency matrix
  4. Summary - Graph

14 Breadth-first search

  1. BFS idea
  2. Using a queue
  3. Marking visited
  4. Summary - BFS

15 Depth-first search

  1. DFS idea
  2. Stack or recursion
  3. Backtracking
  4. Summary - DFS

16 Greedy algorithms

  1. Greedy approach
  2. Coin change
  3. Greedy limits
  4. Summary - Greedy

17 Divide and conquer

  1. Three steps
  2. Recursive structure
  3. Master theorem
  4. Summary - Divide and conquer

18 Backtracking

  1. What is backtracking?
  2. State tree
  3. Pruning
  4. Summary - Backtracking

19 Graph project

  1. Project: Overview
  2. City graph
  3. BFS for reachability
  4. Print the result

20 Intermediate final

  1. Challenge 1
  2. Challenge 2
  3. Challenge 3
  4. Challenge 4