McGee graph

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, the McGee graph or the (3-7)-cage is a 3-regular graph with 24 vertices and 36 edges. The McGee graph is the unique (3,7)-cage (the smallest cubic graph of girth 7). It is also the smallest cubic cage that is not a Moore graph. First discovered by Sachs but unpublished, the graph is named after McGee who published the result in 1960. Then, the McGee graph was proven the unique (3,7)-cage by Tutte in 1966. The McGee graph requires at least eight crossings in any drawing of it in the plane. It is one of three non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these three graphs is the generalized Petersen graph G(12,5), also known as the Nauru graph. The McGee graph has radius 4, diameter 4, chromatic number 3 and chromatic index 3. It is also a 3-vertex-connected and a 3-edge-connected graph. It has book thickness 3 and queue number 2. The graph is 1-planar.

Algebraic properties The characteristic polynomial of the McGee graph is

x 3 ( x − 3 ) ( x − 2 ) 3 ( x + 1 ) 2 ( x + 2 ) ( x 2 + x − 4 ) ( x 3 + x 2 − 4 x − 2 ) 4 {\displaystyle x^{3}(x-3)(x-2)^{3}(x+1)^{2}(x+2)(x^{2}+x-4)(x^{3}+x^{2}-4x-2)^{4}} . The automorphism group of the McGee graph is of order 32 and doesn't act transitively upon its vertices: there are two vertex orbits, of lengths 8 and 16. The McGee graph is the smallest cubic cage that is not a vertex-transitive graph. The automorphism group of the McGee graph, meaning its group of symmetries, has 32 elements. This group is isomorphic to the group of all affine transformations of Z / 8 Z {\displaystyle \mathbb {Z} /8\mathbb {Z} } , i.e., transformations of the form

x ↦ a x + b {\displaystyle x\mapsto ax+b}

where a , b ∈ Z / 8 Z {\displaystyle a,b\in \mathbb {Z} /8\mathbb {Z} } and a {\displaystyle a} is invertible, so a = 1 , 3 , 5 , 7 {\displaystyle a=1,3,5,7} . This is one of the two smallest possible group G {\displaystyle G} with an outer automorphism that maps every element g ∈ G {\displaystyle g\in G} to an element conjugate to g {\displaystyle g} .

Gallery

References

Illustrations

Worked examples

Example 1 — a first encounter with McGee graph

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

In research: McGee graph 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 McGee 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: McGee graph 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 McGee 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 McGee graph in 20 minutes

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

Frequently asked questions

What is McGee graph in simple terms?

In the mathematical field of graph theory, the McGee graph or the (3-7)-cage is a 3-regular graph with 24 vertices and 36 edges. The McGee graph is the unique (3,7)-cage (the smallest cubic graph of girth 7).

Why does McGee graph 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 McGee 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 McGee graph.