Compute graph eigenvalues, spectral radius, energy, and key matrix metrics with clear exports and plots. Turn graph structure into measurable spectral insight fast today.
| Graph | Adjacency Matrix | Main Eigenvalues | Interpretation |
|---|---|---|---|
| Path P3 | [[0,1,0],[1,0,1],[0,1,0]] | -1.4142, 0, 1.4142 | Spectral radius reflects low connectivity. |
| Cycle C4 | [[0,1,0,1],[1,0,1,0],[0,1,0,1],[1,0,1,0]] | -2, 0, 0, 2 | Bipartite symmetry appears around zero. |
| Complete K4 | [[0,1,1,1],[1,0,1,1],[1,1,0,1],[1,1,1,0]] | -1, -1, -1, 3 | Dominant eigenvalue signals dense structure. |
Adjacency spectrum: solve det(A - λI) = 0, where A is the adjacency matrix and λ is an eigenvalue.
Laplacian matrix: L = D - A, where D is the diagonal degree matrix.
Signless Laplacian: Q = D + A.
Normalized Laplacian: ℒ = I - D-1/2 A D-1/2.
Graph energy: E(G) = Σ |λi|.
Spectral radius: ρ(G) = max |λi|.
This page diagonalizes the chosen symmetric graph matrix numerically with the Jacobi rotation method, which is well suited to real symmetric matrices from undirected graphs.
Eigenvalues summarize how a graph distributes connectivity across its vertices. For an adjacency matrix, the largest eigenvalue often rises as density, degree concentration, or weighted influence increases. In practical analysis, this value is used as a compact structural marker because it reacts quickly to added links, hubs, and clustering patterns. Sparse networks usually show smaller dominant values, while dense networks often push the leading eigenvalue upward.
The adjacency spectrum provides more than one headline number. A symmetric spread around zero can indicate bipartite tendencies, while repeated values may hint at regularity or symmetry in the graph design. For complete graphs, one large positive eigenvalue is paired with repeated negative values. For path and cycle graphs, the spectrum is narrower, reflecting simpler topology and fewer high-influence routes between vertices.
Laplacian analysis is especially useful when the goal is to understand cohesion. The smallest Laplacian eigenvalue is always zero, and the second-smallest value, often called algebraic connectivity, measures how strongly the graph remains tied together. If this value is near zero, the network may contain weak bridges or disconnected parts. Higher values usually indicate more robust global connection and improved resistance to fragmentation.
When degrees vary widely, normalized Laplacian results can be more informative than raw adjacency values. Normalization adjusts for high-degree vertices, making comparisons fairer across nodes with very different local influence. This is helpful in transportation, communication, and social graphs where hubs naturally dominate. The normalized spectrum often highlights diffusion, random-walk behavior, and balance between local concentration and wider network accessibility.
Graph energy is the sum of absolute eigenvalues and acts as a compact measure of overall spectral activity. It can rise with complexity, irregularity, or increasing interaction strength. Spectral radius, the largest absolute eigenvalue, is often used in stability and propagation studies because it influences thresholds in network processes. Together, these two outputs help compare candidate graphs without manually inspecting every edge.
This calculator helps students, researchers, and analysts move from a raw matrix to interpretable spectral metrics. The example table supports benchmarking, the exports support reporting, and the chart makes eigenvalue patterns easier to review visually. For best results, enter symmetric matrices for undirected graphs, compare multiple matrix types, and study how spectrum changes when edges are added, removed, or weighted differently.
It is designed for undirected graphs represented by symmetric matrices. Weighted graphs also work if the matrix remains symmetric.
Eigenvalues are defined for square matrices only. Every vertex must have a row and a matching column in the graph matrix.
It gives the largest absolute eigenvalue and is useful for comparing overall graph influence, growth tendency, or propagation sensitivity.
Use Laplacian analysis when connectivity, partitioning, robustness, or algebraic connectivity matters more than raw adjacency influence.
Graph energy is the sum of absolute eigenvalues. It offers a compact numerical view of total spectral activity.
This can happen with empty graphs, isolated vertices, or inputs that contain very weak weights and limited connectivity.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.