science

Hoffman graph

Hoffman 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 Hoffman graph rather than just read about it. In short: In the mathematical field of graph theory, the Hoffman graph is a 4-regular graph with 16 vertices and 32 edges discovered by Alan Hoffman. Published in 1963, it is cospectral to the hypercube graph Q4.

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, the Hoffman graph is a 4-regular graph with 16 vertices and 32 edges discovered by Alan Hoffman. Published in 1963, it is cospectral to the hypercube graph Q4. The Hoffman graph has many common properties with the hypercube Q4—both are Hamiltonian and have chromatic number 2, chromatic index 4, girth 4 and diameter 4. It is also a 4-vertex-connected graph and a 4-edge-connected graph. However, it is not distance-regular and not 1-planar. It has book thickness 3 and queue number 2.

Algebraic properties The Hoffman graph is not a vertex-transitive graph and its full automorphism group is a group of order 48 isomorphic to the direct product of the symmetric group S4 and the cyclic group Z/2Z. Despite not being vertex- or edge-transitive, the Hoffmann graph is still 1-walk-regular (but not distance-regular). The characteristic polynomial of the Hoffman graph is equal to

( x − 4 ) ( x − 2 ) 4 x 6 ( x + 2 ) 4 ( x + 4 ) {\displaystyle (x-4)(x-2)^{4}x^{6}(x+2)^{4}(x+4)}

making it an integral graph—a graph whose spectrum consists entirely of integers. It is the same spectrum as the hypercube Q4.

Gallery

References

Illustrations

Worked examples

Example 1 — a first encounter with Hoffman graph

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

In research: Hoffman 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 Hoffman 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: Hoffman 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 Hoffman 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 Hoffman graph in 20 minutes

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

Frequently asked questions

What is Hoffman graph in simple terms?

Why does Hoffman 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 Hoffman 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 Hoffman graph.