Degree and Handshake Lemma
Sum of degrees is twice the number of edges.
Key idea
Sum of degrees is twice the number of edges.
In depth
The degree of a vertex counts its edges. The sum of all degrees is twice the number of edges, since each edge has two ends. So the number of odd-degree vertices is even.
Worked example
A graph with degrees 3, 3, 2, 2, 2 has 6 edges. It is impossible to have degrees 3, 3, 3.
Formula
Real-world uses
- Checking that a network has an even number of odd-degree links in a sanity check
Common mistake
Forgetting that each edge is counted twice.
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 Degree and Handshake Lemma?
Sum of degrees is twice the number of edges. It is a topic at the advanced level in Discrete Math & Number Theory.
What is the formula for Degree and Handshake Lemma?
The key formula is Σ deg(v) = 2|E|.
Where is Degree and Handshake Lemma used in real life?
Real-world examples: Checking that a network has an even number of odd-degree links in a sanity check.
What should I learn before Degree and Handshake Lemma?
Make sure you are comfortable with Graphs: Basics, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Degree and Handshake Lemma?
Forgetting that each edge is counted twice.
How can I practise Degree and Handshake Lemma?
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 Degree and Handshake Lemma?
Move on to Types of Graphs 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.