science

Hall–Janko graph

Hall–Janko 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 Hall–Janko graph rather than just read about it. In short: In the mathematical field of graph theory, the Hall–Janko graph, also known as the Hall-Janko-Wales graph, is a 36-regular undirected graph with 100 vertices and 1800 edges. It is a rank 3 strongly regular graph with parameters (100,36,14,12) and a maximum coclique of size 10.

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, the Hall–Janko graph, also known as the Hall-Janko-Wales graph, is a 36-regular undirected graph with 100 vertices and 1800 edges. It is a rank 3 strongly regular graph with parameters (100,36,14,12) and a maximum coclique of size 10. This parameter set is not unique, it is however uniquely determined by its parameters as a rank 3 graph. The Hall–Janko graph was originally constructed by D. Wales to establish the existence of the Hall-Janko group as an index 2 subgroup of its automorphism group. The Hall–Janko graph can be constructed out of objects in U3(3), the simple group of order 6048:

In U3(3) there are 36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U3(3) graph. A 168-subgroup has 14 maximal subgroups of order 24, isomorphic to S4. Two 168-subgroups are called adjacent when they intersect in a 24-subgroup. The U3(3) graph is strongly regular, with parameters (36,14,4,6) There are 63 involutions (elements of order 2). A 168-subgroup contains 21 involutions, which are defined to be neighbors. Outside U3(3) let there be a 100th vertex C, whose neighbors are the 36 168-subgroups. A 168-subgroup then has 14 common neighbors with C and in all 1+14+21 neighbors. An involution is found in 12 of the 168-subgroups. C and an involution are non-adjacent, with 12 common neighbors. Two involutions are defined as adjacent when they generate a dihedral subgroup of order 8. An involution has 24 involutions as neighbors. The characteristic polynomial of the Hall–Janko graph is ( x − 36 ) ( x − 6 ) 36 ( x + 4 ) 63 {\displaystyle (x-36)(x-6)^{36}(x+4)^{63}} . Therefore the Hall–Janko graph is an integral graph: its spectrum consists entirely of integers.

References

Illustrations

Worked examples

Example 1 — a first encounter with Hall–Janko graph

Start with the simplest possible case. Write down what Hall–Janko 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 Hall–Janko 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 Hall–Janko 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 Hall–Janko graph

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

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

Frequently asked questions

What is Hall–Janko graph in simple terms?

Why does Hall–Janko 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 Hall–Janko 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 Hall–Janko graph.