Graph Strength Calculator

edge toughness ratio

CalculatorsFreeNo Signup
4.9(722 reviews)
All Tools

Loading tool...

About Graph Strength Calculator

A graph strength calculator computing σ(G) = min |F|/(ω(G-F)-1) over all disconnecting edge sets F. Edge analogue of toughness! σ ≥ 1 necessary for spanning connected subgraph. Tutte-Nash-Williams: σ ≥ k iff k edge-disjoint spanning trees exist. Client-side.

Graph Strength Calculator Features

  • σ(G)
  • |F|/(ω-1)
  • Spanning trees
  • Tutte-NW
  • Common graphs
Graph strength σ(G) = min |F|/(ω(G-F)-1) over disconnecting F. Edge dual of toughness. Beautiful theorem: σ ≥ k ⟺ G has k edge-disjoint spanning trees (Tutte-Nash-Williams). Connects vulnerability to spanning tree packing!

How to Use

Select graph:

  • σ: Strength
  • |F|/(ω-1): Ratio
  • Trees: Packing

Tutte-Nash-Williams

⌊σ(G)⌋ = max number of edge-disjoint spanning trees. Remarkable! Graph strength directly equals spanning tree packing number. One of the most beautiful results in graph theory.

σ vs t

Strength σ: edges removed / (components-1). Toughness t: vertices removed / components. Beautiful duality between edge and vertex vulnerability. σ is computable in polynomial time (via matroid theory); t is NP-hard!

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute σ.
  3. 3Find ⌊σ⌋ spanning trees.
  4. 4Compare with t.
  5. 5Apply Tutte-NW.

Graph Strength Calculator — Frequently Asked Questions

Why is strength so special?+

Unlike toughness (NP-hard), strength is polynomial-time computable via matroid intersection. AND it directly gives the spanning tree packing number. A rare case where a vulnerability parameter is both meaningful and tractable.

What's the Tutte-Nash-Williams theorem?+

Graph G has k edge-disjoint spanning trees ⟺ for every partition P of V, sum of edges crossing parts ≥ k(|P|-1). Equivalently: ⌊σ(G)⌋ = max spanning tree packing. Proved independently by Tutte and Nash-Williams.

How does strength relate to connectivity?+

λ(G) ≤ 2σ(G). Strength provides a more refined measure. High strength means the graph has many edge-disjoint spanning trees, implying high redundancy.

More Calculators Tools

Explore all 639 other calculators tools available on Generatr

1-Sample Z-Test Calculator2-Sample Z-Test Calculator3D Surface Area CalculatorA/B Testing Significance CalculatorAbsolute Value CalculatorAbundant Number CheckerAccelerated Aging CalculatorAchromatic Number CalculatorAcyclic Chromatic Number CalculatorAdditive Persistence CalculatorAge CalculatorAging Test CalculatorAlbertson Index CalculatorAlgebraic Connectivity CalculatorAliquot Sequence CalculatorAliquot Sum CalculatorAlternating Series Test CalculatorAmicable Number FinderAnnuity CalculatorAPY CalculatorAquarium CO2 Drop CheckerArea CalculatorArea Under Curve CalculatorArithmetic Sequence CalculatorArmstrong Number CalculatorASCVD Risk CalculatorAspect Ratio CalculatorAtom Bond Connectivity CalculatorAugmented Eccentric Connectivity CalculatorAugmented Zagreb Index Calculator
View all 639 calculators tools
Share this tool: