mathematics

Rank (graph theory)

Rank (graph theory) is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Rank (graph theory) rather than just read about it. In short: In graph theory, a branch of mathematics, the rank of an undirected graph has two unrelated definitions. Let n equal the number of vertices of the graph.

Key takeaways

Rank (graph theory) belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rank (graph theory) to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rank (graph theory) from memory before moving on to harder problems.

Reference excerpt

In graph theory, a branch of mathematics, the rank of an undirected graph has two unrelated definitions. Let n equal the number of vertices of the graph.

In the matrix theory of graphs the rank r of an undirected graph is defined as the rank of its adjacency matrix. Analogously, the nullity of the graph is the nullity of its adjacency matrix, which equals n − r. In the matroid theory of graphs the rank of an undirected graph is defined as the number n − c, where c is the number of connected components of the graph. Equivalently, the rank of a graph is the rank of the oriented incidence matrix associated with the graph. Analogously, the nullity of the graph is the nullity of its oriented incidence matrix, given by the formula m − n + c, where n and c are as above and m is the number of edges in the graph. The nullity is equal to the first Betti number of the graph. The sum of the rank and the nullity is the number of edges.

Examples A sample graph and matrix:

(corresponding to the four edges, e1–e4):

In this example, the matrix theory rank of the matrix is 4, because its column vectors are linearly independent.

See also Circuit rank Cycle rank Nullity (graph theory)

Notes

References Chen, Wai-Kai (1976), Applied Graph Theory, North Holland Publishing Company, ISBN 0-7204-2371-6. Hedetniemi, S. T., Jacobs, D. P., Laskar, R. (1989), Inequalities involving the rank of a graph. Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 6, pp. 173–176. Bevis, Jean H., Blount, Kevin K., Davis, George J., Domke, Gayla S., Miller, Valerie A. (1997), The rank of a graph after vertex addition. Linear Algebra and its Applications, vol. 265, pp. 55–69.

Worked examples

Example 1 — a first encounter with Rank (graph theory)

Start with the simplest possible case. Write down what Rank (graph theory) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rank (graph theory) before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rank (graph theory) ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rank (graph theory)

In research: Rank (graph theory) appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rank (graph theory) in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rank (graph theory) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic graph theory, Graph connectivity, Graph invariants, so understanding it makes those chapters shorter.
In everyday life: Look for Rank (graph theory) outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Rank (graph theory)” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Rank (graph theory) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rank (graph theory) means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Rank (graph theory) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rank (graph theory) in simple terms?

In graph theory, a branch of mathematics, the rank of an undirected graph has two unrelated definitions. Let n equal the number of vertices of the graph.

Why does Rank (graph theory) matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rank (graph theory)?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rank (graph theory).