chemistry

Padmakar–Ivan index

Padmakar–Ivan index is a chemistry 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 Padmakar–Ivan index rather than just read about it. In short: In chemical graph theory, the Padmakar–Ivan (PI) index is a topological index of a molecule, used in biochemistry. The Padmakar–Ivan index is a generalization introduced by Padmakar V.

Key takeaways

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

Reference excerpt

In chemical graph theory, the Padmakar–Ivan (PI) index is a topological index of a molecule, used in biochemistry. The Padmakar–Ivan index is a generalization introduced by Padmakar V. Khadikar and Iván Gutman of the concept of the Wiener index, introduced by Harry Wiener. The Padmakar–Ivan index of a graph G is the sum over all edges uv of G of number of edges which are not equidistant from u and v. Let G be a graph and e = uv an edge of G. Here n e u ( e ∣ G ) {\displaystyle n_{eu}(e\mid G)} denotes the number of edges lying closer to the vertex u than the vertex v, and n e v ( e ∣ G ) {\displaystyle n_{ev}(e\mid G)} is the number of edges lying closer to the vertex v than the vertex u. The Padmakar–Ivan index of a graph G is defined as

PI ⁡ ( G ) = ∑ e ∈ E ( G ) [ n e u ( e ∣ G ) + n e v ( e ∣ G ) ] {\displaystyle \operatorname {PI} (G)=\sum _{e\in E(G)}[n_{eu}(e\mid G)+n_{ev}(e\mid G)]}

The PI index is very important in the study of quantitative structure–activity relationship for the classification models used in the chemical, biological sciences, engineering, and nanotechnology.

Examples

The PI index of Dendrimer Nanostar of the following figure can be calculated by

PI ⁡ ( G n ) = 441 ⋅ 4 n − 639 ⋅ 2 n + 232 , n ≥ 0. {\displaystyle \operatorname {PI} (G_{n})=441\cdot 4^{n}-639\cdot 2^{n}+232,\quad n\geq 0.}

The double graph of a graph G {\displaystyle G} , denoted D [ G ] {\displaystyle {\mathcal {D}}[G]} , has a known index in relation to G {\displaystyle G} itself:

PI ⁡ ( D [ G ] ) = 8 PI ⁡ ( G ) {\displaystyle \operatorname {PI} ({\mathcal {D}}[G])=8\operatorname {PI} (G)}

References

Worked examples

Example 1 — a first encounter with Padmakar–Ivan index

Start with the simplest possible case. Write down what Padmakar–Ivan 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 Padmakar–Ivan 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 Padmakar–Ivan 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 Padmakar–Ivan index

In research: Padmakar–Ivan 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 Padmakar–Ivan 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: Padmakar–Ivan 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 Padmakar–Ivan 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.

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How to study Padmakar–Ivan index in 20 minutes

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

Frequently asked questions

What is Padmakar–Ivan index in simple terms?

In chemical graph theory, the Padmakar–Ivan (PI) index is a topological index of a molecule, used in biochemistry. The Padmakar–Ivan index is a generalization introduced by Padmakar V.

Why does Padmakar–Ivan 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 Padmakar–Ivan 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 Padmakar–Ivan index.