Robertson graph

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson. The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964. As a cage graph, it is the smallest 4-regular graph with girth 5. It has chromatic number 3, chromatic index 5, diameter 3, radius 3 and is both 4-vertex-connected and 4-edge-connected. It has book thickness 3 and queue number 2. The graph is neither planar nor 1-planar. The Robertson graph is also a Hamiltonian graph which possesses 5,376 distinct directed Hamiltonian cycles. The Robertson graph is one of the smallest graphs with cop number 4.

Algebraic properties The Robertson graph is not a vertex-transitive graph; its full automorphism group is isomorphic to the dihedral group of order 24, the group of symmetries of a regular dodecagon, including both rotations and reflections. The characteristic polynomial of the Robertson graph is

( x − 4 ) ( x − 1 ) 2 ( x 2 − 3 ) 2 ( x 2 + x − 5 ) {\displaystyle (x-4)(x-1)^{2}(x^{2}-3)^{2}(x^{2}+x-5)}

( x 2 + x − 4 ) 2 ( x 2 + x − 3 ) 2 ( x 2 + x − 1 ) . {\displaystyle (x^{2}+x-4)^{2}(x^{2}+x-3)^{2}(x^{2}+x-1).\ }

Gallery

References

Illustrations

Worked examples

Example 1 — a first encounter with Robertson graph

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

In research: Robertson 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 Robertson 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: Robertson 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 Robertson 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 Robertson graph in 20 minutes

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

Frequently asked questions

What is Robertson graph in simple terms?

In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson. The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964.

Why does Robertson 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 Robertson 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 Robertson graph.