science

Gewirtz graph

Gewirtz 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 Gewirtz graph rather than just read about it. In short: The Gewirtz graph is a strongly regular graph with 56 vertices and valency 10. It is named after the mathematician Allan Gewirtz, who described the graph in his dissertation.

Key takeaways

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

Reference excerpt

The Gewirtz graph is a strongly regular graph with 56 vertices and valency 10. It is named after the mathematician Allan Gewirtz, who described the graph in his dissertation.

Construction The Gewirtz graph can be constructed as follows. Consider the unique S(3, 6, 22) Steiner system, with 22 elements and 77 blocks. Choose an arbitrary element, and let the vertices be the 56 blocks not containing it. Two blocks are adjacent when they are disjoint. With this construction, one can embed the Gewirtz graph in the Higman–Sims graph.

Properties The characteristic polynomial of the Gewirtz graph is

( x − 10 ) ( x − 2 ) 35 ( x + 4 ) 20 . {\displaystyle (x-10)(x-2)^{35}(x+4)^{20}.\,}

Therefore, it is an integral graph. The Gewirtz graph is also determined by its spectrum. The independence number is 16.

Notes

References Brouwer, Andries. "Sims-Gewirtz graph". Weisstein, Eric W. "Gewirtz graph". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Gewirtz graph

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

In research: Gewirtz 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 Gewirtz 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: Gewirtz graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Individual graphs, Regular graphs, Strongly regular graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Gewirtz 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 Gewirtz graph in 20 minutes

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

Frequently asked questions

What is Gewirtz graph in simple terms?

The Gewirtz graph is a strongly regular graph with 56 vertices and valency 10. It is named after the mathematician Allan Gewirtz, who described the graph in his dissertation.

Why does Gewirtz 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 Gewirtz 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 Gewirtz graph.