Adjacency Matrices
Represent graphs as matrices.
Key idea
Represent graphs as matrices.
In depth
An adjacency matrix A has Aᵢⱼ = 1 if i and j are adjacent. The entry (Aᵏ)ᵢⱼ counts walks of length k from i to j. Symmetric for undirected graphs.
Worked example
For a triangle, A³ has diagonal entries 2, since there are 2 closed walks of length 3 from each vertex.
Real-world uses
- Storing a network in a computer and computing how many paths join two nodes
Common mistake
Mixing up walks with paths when interpreting powers.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Adjacency Matrices?
Represent graphs as matrices. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Adjacency Matrices used in real life?
Real-world examples: Storing a network in a computer and computing how many paths join two nodes.
What should I learn before Adjacency Matrices?
Make sure you are comfortable with Network Flow and Max-Flow Min-Cut, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Adjacency Matrices?
Mixing up walks with paths when interpreting powers.
How can I practise Adjacency Matrices?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Adjacency Matrices?
Move on to Divisibility next.
Related advanced topics in Discrete Math & Number Theory
- Tautologies and Contradictions – Always true or always false statements.
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.