Padmakar Ivan Index Calculator

edge-partition asymmetry

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About Padmakar Ivan Index Calculator

A Padmakar-Ivan index calculator computing PI(G) = Σ_{e=(i,j)} |n_i(e) - n_j(e)| where n_i(e) = vertices closer to i. Khadikar (2001). PI = 0 iff vertex-transitive. Measures asymmetry of edge-partitioning. Complements Szeged index. Client-side.

Padmakar Ivan Index Calculator Features

  • PI(G)
  • |nᵢ-nⱼ|
  • Asymmetry
  • Khadikar
  • Common graphs
Padmakar-Ivan PI(G) = Σ |nᵢ(e)-nⱼ(e)|. For each edge: how asymmetrically does it divide the graph? PI = 0 for vertex-transitive (symmetric) graphs. High PI = highly asymmetric edge partitions. Khadikar (2001).

How to Use

Select graph:

  • PI: PI index
  • |n-n|: Per edge
  • Sym: Symmetry

Symmetry Detection

PI = 0 ⟺ every edge divides the graph equally (vertex-transitive). K_n: PI=0. C_n: PI=0 (even). The closer PI is to 0, the more 'symmetric' the graph. Instant symmetry test!

PI vs Szeged

Szeged: Σ nᵢ·nⱼ (product). PI: Σ |nᵢ-nⱼ| (difference). Szeged captures total partition quality. PI captures partition imbalance. They give complementary information.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2For each edge: |nᵢ-nⱼ|.
  3. 3Sum all terms.
  4. 4Check if PI=0.
  5. 5Assess symmetry.

Padmakar Ivan Index Calculator — Frequently Asked Questions

When is PI = 0?+

PI = 0 ⟺ every edge has nᵢ(e) = nⱼ(e). This means every edge divides the graph into equal halves. Happens for vertex-transitive graphs: K_n, C_{2k}, Petersen, etc.

What does high PI mean?+

High PI = many edges divide the graph very asymmetrically. Stars have high PI (hub-leaf edges: n₁≈n-1, n₂≈1, difference ≈ n-2). Asymmetric, unbalanced structures.

Connection to Mostar index?+

Mostar index = Σ |nᵢ-nⱼ| over edges (exactly PI!). They're the same invariant, introduced independently. Mostar name came from the famous bridge in Bosnia — edges as 'bridges' between two sides.

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