science

Perkel graph

Perkel 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 Perkel graph rather than just read about it. In short: In mathematics, the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3).

Key takeaways

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

Reference excerpt

In mathematics, the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3). The Perkel graph is also distance-transitive. It is also the skeleton of an abstract regular polytope, the 57-cell. The vertex set is Z3 × Z19 where (i,j) is joined to (i+1,k) when (k−j)3 = 26i.

References

Brouwer, A. E. Perkel Graph. [1]. Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. The Perkel Graph for L(2,19). 13.3 in Distance Regular Graphs. New York: Springer-Verlag, pp. 401–403, 1989. Perkel, M. Bounding the Valency of Polygonal Graphs with Odd Girth. Can. J. Math. 31, 1307-1321, 1979. Perkel, M. Characterization of in Terms of Its Geometry.Geom. Dedicata 9, 291-298, 1980.

External links Weisstein, Eric W. "Perkel graph". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Perkel graph

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

In research: Perkel 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 Perkel 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: Perkel 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 Perkel 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 Perkel graph in 20 minutes

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

Frequently asked questions

What is Perkel graph in simple terms?

In mathematics, the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3).

Why does Perkel 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 Perkel 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 Perkel graph.