science

Harries graph

Harries graph 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 Harries graph rather than just read about it. In short: In the mathematical field of graph theory, the Harries graph or Harries (3-10)-cage is a 3-regular, undirected graph with 70 vertices and 105 edges. The Harries graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian.

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, the Harries graph or Harries (3-10)-cage is a 3-regular, undirected graph with 70 vertices and 105 edges. The Harries graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian. It is also a 3-vertex-connected and 3-edge-connected, non-planar, cubic graph. It has book thickness 3 and queue number 2. The characteristic polynomial of the Harries graph is

( x − 3 ) ( x − 1 ) 4 ( x + 1 ) 4 ( x + 3 ) ( x 2 − 6 ) ( x 2 − 2 ) ( x 4 − 6 x 2 + 2 ) 5 ( x 4 − 6 x 2 + 3 ) 4 ( x 4 − 6 x 2 + 6 ) 5 . {\displaystyle (x-3)(x-1)^{4}(x+1)^{4}(x+3)(x^{2}-6)(x^{2}-2)(x^{4}-6x^{2}+2)^{5}(x^{4}-6x^{2}+3)^{4}(x^{4}-6x^{2}+6)^{5}.\,}

History In 1972, A. T. Balaban published a (3-10)-cage graph, a cubic graph that has as few vertices as possible for girth 10. It was the first (3-10)-cage discovered but it was not unique. The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong in 1980. There exist three distinct (3-10)-cage graphs—the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Moreover, the Harries–Wong graph and Harries graph are cospectral graphs.

Gallery

References

Illustrations

Worked examples

Example 1 — a first encounter with Harries graph

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

In research: Harries 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 Harries 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: Harries 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 Harries 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 Harries graph in 20 minutes

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

Frequently asked questions

What is Harries graph in simple terms?

Why does Harries 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 Harries 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 Harries graph.