science

Holt graph

Holt 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 Holt graph rather than just read about it. In short: In graph theory, the Holt graph or Doyle graph is the smallest half-transitive graph, that is, the smallest example of a vertex-transitive and edge-transitive graph which is not also symmetric. Such graphs are not common.

Key takeaways

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

Reference excerpt

In graph theory, the Holt graph or Doyle graph is the smallest half-transitive graph, that is, the smallest example of a vertex-transitive and edge-transitive graph which is not also symmetric. Such graphs are not common. It is named after Peter G. Doyle and Derek F. Holt, who discovered the same graph independently in 1976 and 1981 respectively. The Holt graph has diameter 3, radius 3 and girth 5, chromatic number 3, chromatic index 5 and is Hamiltonian with 98,472 distinct Hamiltonian cycles. It is also a 4-vertex-connected and a 4-edge-connected graph. It has book thickness 3 and queue number 3. The graph is not 1-planar. It has an automorphism group of order 54. This is a smaller group than a symmetric graph with the same number of vertices and edges would have. The graph drawing on the right highlights this, in that it lacks reflectional symmetry. The characteristic polynomial of the Holt graph is

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

Gallery

References

Illustrations

Worked examples

Example 1 — a first encounter with Holt graph

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

In research: Holt 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 Holt 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: Holt 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 Holt 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Holt graph” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Holt graph in 20 minutes

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

Frequently asked questions

What is Holt graph in simple terms?

Why does Holt 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 Holt 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 Holt graph.