mathematics

Tutte–Coxeter graph

Tutte–Coxeter graph 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 Tutte–Coxeter graph rather than just read about it. In short: In the mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth 8, it is a cage and a Moore graph.

Key takeaways

Tutte–Coxeter 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 Tutte–Coxeter graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tutte–Coxeter graph from memory before moving on to harder problems.

Reference excerpt

In the mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth 8, it is a cage and a Moore graph. It is bipartite, and can be constructed as the Levi graph of the generalized quadrangle W2 (known as the Cremona–Richmond configuration). The graph is named after William Thomas Tutte and H. S. M. Coxeter; it was discovered by Tutte (1947) but its connection to geometric configurations was investigated by both authors in a pair of jointly published papers (Tutte 1958; Coxeter 1958a). All the cubic distance-regular graphs are known. The Tutte–Coxeter is one of the 13 such graphs. It has crossing number 13, book thickness 3 and queue number 2.

Constructions and automorphisms The Tutte–Coxeter graph is the bipartite Levi graph connecting the 15 perfect matchings of a 6-vertex complete graph K6 to its 15 edges, as described by Coxeter (1958b), based on work by Sylvester (1844). Each vertex corresponds to an edge or a perfect matching, and connected vertices represent the incidence structure between edges and matchings. Based on this construction, Coxeter showed that the Tutte–Coxeter graph is a symmetric graph; it has a group of 1440 automorphisms, which may be identified with the automorphisms of the group of permutations on six elements (Coxeter 1958b). The inner automorphisms of this group correspond to permuting the six vertices of the K6 graph; these permutations act on the Tutte–Coxeter graph by permuting the vertices on each side of its bipartition while keeping each of the two sides fixed as a set. In addition, the outer automorphisms of the group of permutations swap one side of the bipartition for the other. As Coxeter showed, any path of up to five edges in the Tutte–Coxeter graph is equivalent to any other such path by one such automorphism.

The Tutte–Coxeter graph as a building This graph is the spherical building associated to the symplectic group S p 4 ( F 2 ) {\displaystyle Sp_{4}(\mathbb {F} _{2})} (there is an exceptional isomorphism between this group and the symmetric group S 6 {\displaystyle S_{6}} ). More specifically, it is the incidence graph of a generalized quadrangle. Concretely, the Tutte-Coxeter graph can be defined from a 4-dimensional symplectic vector space V {\displaystyle V} over F 2 {\displaystyle \mathbb {F} _{2}} as follows:

vertices are either nonzero vectors, or isotropic 2-dimensional subspaces, there is an edge between a nonzero vector v and an isotropic 2-dimensional subspace W ⊂ V {\displaystyle W\subset V} if and only if v ∈ W {\displaystyle v\in W} .

Gallery

References

Coxeter, H. S. M. (1958a). "The chords of the non-ruled quadric in PG(3,3)". Can. J. Math. 10: 484–488. doi:10.4153/CJM-1958-047-0. Coxeter, H. S. M. (1958b). "Twelve points in PG(5,3) with 95040 self-transformations". Proceedings of the Royal Society A. 247 (1250): 279–293. Bibcode:1958RSPSA.247..279C. doi:10.1098/rspa.1958.0184. JSTOR 100667. S2CID 121676627. Sylvester, J. J. (1844). "Elementary researches in the analysis of combinatorial aggregation". Phil. Mag. Series 3. 24: 285–295. doi:10.1080/14786444408644856. Tutte, W. T. (1947). "A family of cubical graphs". Proc. Cambridge Philos. Soc. 43 (4): 459–474. Bibcode:1947PCPS...43..459T. doi:10.1017/S0305004100023720. S2CID 123505185. Tutte, W. T. (1958). "The chords of the non-ruled quadric in PG(3,3)". Can. J. Math. 10: 481–483. doi:10.4153/CJM-1958-046-3.

External links François Labelle. "3D Model of Tutte's 8-cage". Weisstein, Eric W. "Levi Graph". MathWorld. Exoo, G. "Rectilinear Drawings of Famous Graphs." [1]

Illustrations

Worked examples

Example 1 — a first encounter with Tutte–Coxeter graph

Start with the simplest possible case. Write down what Tutte–Coxeter 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 Tutte–Coxeter 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 Tutte–Coxeter 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 Tutte–Coxeter graph

In research: Tutte–Coxeter 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 Tutte–Coxeter 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: Tutte–Coxeter graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1958 introductions, Configurations (geometry), Individual graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Tutte–Coxeter 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.

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How to study Tutte–Coxeter graph in 20 minutes

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

Frequently asked questions

What is Tutte–Coxeter graph in simple terms?

Why does Tutte–Coxeter 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 Tutte–Coxeter 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 Tutte–Coxeter graph.