Hyper-Wiener index

Key takeaways

Hyper-Wiener index belongs to chemistry; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hyper-Wiener index to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hyper-Wiener index from memory before moving on to harder problems.

Reference excerpt

In chemical graph theory, the hyper-Wiener index or hyper-Wiener number is a topological index of a molecule, used in biochemistry. The hyper-Wiener index is a generalization introduced by Milan Randić of the concept of the Wiener index, introduced by Harry Wiener. The hyper-Wiener index of a connected graph G is defined by

W W ( G ) = 1 2 ∑ u , v ∈ V ( G ) ( d ( u , v ) + d 2 ( u , v ) ) , {\displaystyle WW(G)={\frac {1}{2}}\sum _{u,v\in V(G)}(d(u,v)+d^{2}(u,v)),}

where d(u,v) is the distance between vertex u and v. Hyper-Wiener index as topological index assigned to G = (V,E) is based on the distance function which is invariant under the action of the automorphism group of G. Hyper-Wiener index can be used for the representation of computer networks and enhancing lattice hardware security. Hyper-Wiener indices used to limit the structure of a particle into a solitary number which signifies the sub-atomic stretching and electronic structures.

Example One-pentagonal carbon nanocone which is an infinite symmetric graph, consists of one pentagon as its core surrounded by layers of hexagons. If there are n layers, then the graph of the molecules is denoted by Gn. we have the following explicit formula for hyper-Wiener index of one-pentagonal carbon nanocone,

WW ⁡ ( G n ) = 20 + 533 4 n + 8501 24 n 2 + 5795 12 n 3 + 8575 24 n 4 + 409 3 n 5 + 21 n 6 {\displaystyle \operatorname {WW} (G_{n})=20+{\frac {533}{4}}n+{\frac {8501}{24}}n^{2}+{\frac {5795}{12}}n^{3}+{\frac {8575}{24}}n^{4}+{\frac {409}{3}}n^{5}+21n^{6}}

References

Worked examples

Example 1 — a first encounter with Hyper-Wiener index

Start with the simplest possible case. Write down what Hyper-Wiener index claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In chemistry, 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 Hyper-Wiener index 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 Hyper-Wiener index 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 Hyper-Wiener index

In research: Hyper-Wiener index appears in chemistry 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 Hyper-Wiener index 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: Hyper-Wiener index is common in secondary-school and first-year university syllabi. It links to neighbouring topics Cheminformatics, Graph invariants, Mathematical chemistry, so understanding it makes those chapters shorter.
In everyday life: Look for Hyper-Wiener index 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.

How to study Hyper-Wiener index in 20 minutes

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

Frequently asked questions

What is Hyper-Wiener index in simple terms?

In chemical graph theory, the hyper-Wiener index or hyper-Wiener number is a topological index of a molecule, used in biochemistry. The hyper-Wiener index is a generalization introduced by Milan Randić of the concept of the Wiener index, introduced by Harry Wiener.

Why does Hyper-Wiener index matter?

Because it connects several chemistry 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 Hyper-Wiener index?

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 Hyper-Wiener index.