Ramanujan graph

Key takeaways

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

Reference excerpt

In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. As Murty's survey paper notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". These graphs are indirectly named after Srinivasa Ramanujan; their name comes from the Ramanujan–Petersson conjecture, which was used in a construction of some of these graphs.

Definition Let G {\displaystyle G} be a connected d {\displaystyle d} -regular graph with n {\displaystyle n} vertices, and let λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}} be the eigenvalues of the adjacency matrix of G {\displaystyle G} (or the spectrum of G {\displaystyle G} ). Because G {\displaystyle G} is connected and d {\displaystyle d} -regular, its eigenvalues satisfy d = λ 1 > λ 2 {\displaystyle d=\lambda _{1}>\lambda _{2}} ≥ ⋯ ≥ λ n ≥ − d {\displaystyle \geq \cdots \geq \lambda _{n}\geq -d} . Define λ ( G ) = max i ≠ 1 | λ i | = max ( | λ 2 | , … , | λ n | ) {\displaystyle \lambda (G)=\max _{i\neq 1}|\lambda _{i}|=\max(|\lambda _{2}|,\ldots ,|\lambda _{n}|)} . A connected d {\displaystyle d} -regular graph G {\displaystyle G} is a Ramanujan graph if λ ( G ) ≤ 2 d − 1 {\displaystyle \lambda (G)\leq 2{\sqrt {d-1}}} . Many sources uses an alternative definition λ ′ ( G ) = max | λ i | < d | λ i | {\displaystyle \lambda '(G)=\max _{|\lambda _{i}|<d}|\lambda _{i}|} (whenever there exists λ i {\displaystyle \lambda _{i}} with | λ i | < d {\displaystyle |\lambda _{i}|<d} ) to define Ramanujan graphs. In other words, we allow − d {\displaystyle -d} in addition to the "small" eigenvalues. Since λ n = − d {\displaystyle \lambda _{n}=-d} if and only if the graph is bipartite, we will refer to the graphs that satisfy this alternative definition but not the first definition as bipartite Ramanujan graphs. If G {\displaystyle G} is a Ramanujan graph, then G × K 2 {\displaystyle G\times K_{2}} is a bipartite Ramanujan graph, so the existence of Ramanujan graphs is stronger. As observed by Toshikazu Sunada, a regular graph is Ramanujan if and only if its Ihara zeta function satisfies an analog of the Riemann hypothesis.

Examples and constructions

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Worked examples

Example 1 — a first encounter with Ramanujan graph

Start with the simplest possible case. Write down what Ramanujan graph 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 Ramanujan graph 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 Ramanujan graph 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 Ramanujan graph

In research: Ramanujan graph 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 Ramanujan graph 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: Ramanujan graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic graph theory, Graph families, Regular graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Ramanujan graph 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.

How to study Ramanujan graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ramanujan graph 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 Ramanujan graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ramanujan graph in simple terms?

In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders.

Why does Ramanujan graph 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 Ramanujan graph?

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 Ramanujan graph.