Calculator
How to use this calculator
- Choose n, then select Adjacency matrix or Edge list.
- Paste your graph data. For edge lists, set node indexing and optional weights.
- Enable Symmetrize if your data is directed or asymmetric.
- Select the Laplacian type and precision, then press Calculate.
- Review λ₂, eigenvalues, and the Fiedler vector. Use exports for reports.
Example data table
Sample undirected graph with 5 nodes. Expected λ₂ is about 0.3820 for this path-like structure.
| Example | Input mode | n | Graph input | Suggested options | Typical λ₂ |
|---|---|---|---|---|---|
| Path-ish (5 nodes) | Adjacency matrix | 5 | 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 |
Unweighted, Symmetrize ON | ~0.3820 |
| Cycle C5 | Edge list | 5 | 1 2 2 3 3 4 4 5 5 1 |
Unweighted, Symmetrize ON | ~1.3820 |
| Weighted triangle + tail | Edge list | 4 | 1 2 2.0 2 3 1.5 1 3 1.0 3 4 0.4 |
Weighted ON, Symmetrize ON | Depends on weights |
Algebraic connectivity in practice
1) What algebraic connectivity measures
Algebraic connectivity is the second-smallest eigenvalue, λ2, of a graph Laplacian. It quantifies how tightly the network holds together: larger λ2 usually means fewer bottlenecks, better diffusion, and faster consensus. When λ2 is near zero, the graph is disconnected or barely linked by weak bridges.
2) Laplacian choice and expected ranges
With the combinatorial Laplacian (D − A), λ2 depends on overall degree scale. With the normalized form, eigenvalues lie between 0 and 2, which helps comparisons across graphs with different densities. For normalized Laplacians, λ2 approaching 2 indicates very strong global coupling.
3) Interpreting λ2 using simple examples
For a path of n nodes, λ2 is small and shrinks as n grows, reflecting vulnerability to cuts. Cycles typically have higher λ2 than paths of the same size because there is no single critical bridge. Complete graphs produce large λ2 because every node connects broadly.
4) Weighted graphs and physical meaning
In weighted networks, edge weights act like coupling strengths, conductances, or interaction rates. Increasing a weight on a bottleneck edge can raise λ2 noticeably, improving robustness and reducing effective separation between clusters. Negative weights are uncommon for connectivity studies and can distort interpretation.
5) The Fiedler vector for partitioning
The eigenvector associated with λ2 is the Fiedler vector. Sorting nodes by its values often reveals a natural split: nodes with positive entries tend to group together, and nodes with negative entries form another group. This is a practical spectral method for finding approximate minimum cuts.
6) Sensitivity to data preparation
Algebraic connectivity is defined for undirected graphs, so asymmetric inputs should be symmetrized. Self-loops usually do not help connectivity and are commonly ignored. Isolated nodes force λ2 to be zero, even if the remaining subgraph is well connected.
7) Numerical stability and scaling
Spectral calculations are sensitive to rounding for very small λ2. Using more display precision helps distinguish “almost zero” from truly zero. For large n, exact eigen-decomposition can be costly; in applications, iterative methods are often used to estimate only the smallest few eigenvalues.
8) How to use results for decisions
Use λ2 to compare design alternatives: add or strengthen edges where the Fiedler vector suggests a cut, test how removing nodes changes λ2, and choose normalized Laplacians when graph density varies. These steps support robust network design and reliable transport behavior.
FAQs
1) What does λ2 = 0 mean?
It usually means the graph is disconnected, or it contains isolated nodes. Multiple zero eigenvalues indicate multiple connected components in the undirected interpretation.
2) Why is algebraic connectivity called the Fiedler value?
It is named after Miroslav Fiedler, who studied how the second-smallest Laplacian eigenvalue relates to connectivity and partitioning in graphs.
3) Should I use combinatorial or normalized Laplacian?
Use combinatorial when degree scale is meaningful in your model. Use normalized when you want comparisons across graphs with different densities or degree variation.
4) Can λ2 be negative?
For a proper undirected graph with nonnegative weights, Laplacian eigenvalues are nonnegative. Negative values typically signal non-symmetric input, negative weights, or numerical issues.
5) How do weights affect λ2?
Stronger weights on edges increase coupling and can raise λ2, especially if they reinforce weak bridges. Weights on already dense regions may change λ2 less.
6) What can I do with the Fiedler vector?
Use it to suggest a partition: split nodes by sign or by a threshold on the vector values. This often identifies bottlenecks and near-minimum cuts.
7) Why does symmetrizing matter?
Algebraic connectivity assumes an undirected Laplacian. Symmetrizing converts asymmetric data into an undirected representation, preventing misleading eigenvalues and making λ2 interpretable.