science

Tutte 12-cage

Tutte 12-cage is a science 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 Tutte 12-cage rather than just read about it. In short: In the mathematical field of graph theory, the Tutte 12-cage or Benson graph is a 3-regular graph with 126 vertices and 189 edges. It is named after W.

Key takeaways

Tutte 12-cage belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Tutte 12-cage to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tutte 12-cage from memory before moving on to harder problems.

Reference excerpt

In the mathematical field of graph theory, the Tutte 12-cage or Benson graph is a 3-regular graph with 126 vertices and 189 edges. It is named after W. T. Tutte. The Tutte 12-cage is the unique (3-12)-cage (sequence A052453 in the OEIS). It was discovered by C. T. Benson in 1966. It has chromatic number 2 (bipartite), chromatic index 3, girth 12 (as a 12-cage) and diameter 6. Its crossing number is known to be less than 165, see Wolfram MathWorld.

Construction The Tutte 12-cage is a cubic Hamiltonian graph and can be defined by the LCF notation [17, 27, −13, −59, −35, 35, −11, 13, −53, 53, −27, 21, 57, 11, −21, −57, 59, −17]7. There are, up to isomorphism, precisely two generalized hexagons of order (2,2) as proved by Cohen and Tits. They are the split Cayley hexagon H(2) and its point-line dual. Clearly both of them have the same incidence graph, which is in fact isomorphic to the Tutte 12-cage. The Balaban 11-cage can be constructed by excision from the Tutte 12-cage by removing a small subtree and suppressing the resulting vertices of degree two.

Algebraic properties The automorphism group of the Tutte 12-cage is of order 12,096 and is a semi-direct product of the projective special unitary group PSU(3,3) with the cyclic group Z/2Z. It acts transitively on its edges but not on its vertices, making it a semi-symmetric graph, a regular graph that is edge-transitive but not vertex-transitive. In fact, the automorphism group of the Tutte 12-cage preserves the bipartite parts and acts primitively on each part. Such graphs are called bi-primitive graphs and only five cubic bi-primitive graphs exist; they are named the Iofinova-Ivanov graphs and are of order 110, 126, 182, 506 and 990. All the cubic semi-symmetric graphs on up to 768 vertices are known. According to Conder, Malnič, Marušič and Potočnik, the Tutte 12-cage is the unique cubic semi-symmetric graph on 126 vertices and is the fifth smallest possible cubic semi-symmetric graph after the Gray graph, the Iofinova–Ivanov graph on 110 vertices, the Ljubljana graph and a graph on 120 vertices with girth 8. The characteristic polynomial of the Tutte 12-cage is

( x − 3 ) x 28 ( x + 3 ) ( x 2 − 6 ) 21 ( x 2 − 2 ) 27 . {\displaystyle (x-3)x^{28}(x+3)(x^{2}-6)^{21}(x^{2}-2)^{27}.\ }

It is the only graph with this characteristic polynomial; therefore, the 12-cage is determined by its spectrum.

Gallery

References

Illustrations

Worked examples

Example 1 — a first encounter with Tutte 12-cage

Start with the simplest possible case. Write down what Tutte 12-cage claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Tutte 12-cage 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 Tutte 12-cage 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 Tutte 12-cage

In research: Tutte 12-cage appears in science 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 Tutte 12-cage 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: Tutte 12-cage is common in secondary-school and first-year university syllabi. It links to neighbouring topics Individual graphs, Regular graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Tutte 12-cage 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.

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How to study Tutte 12-cage in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Tutte 12-cage 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 Tutte 12-cage out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Tutte 12-cage in simple terms?

In the mathematical field of graph theory, the Tutte 12-cage or Benson graph is a 3-regular graph with 126 vertices and 189 edges. It is named after W.

Why does Tutte 12-cage matter?

Because it connects several science 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 Tutte 12-cage?

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 Tutte 12-cage.