1. Choose a graph
Vertices are labeled $1,2,\ldots,n$. Click two vertices in the drawing to toggle an edge.
2. Graph drawing
Node color shows the Fiedler vector when the graph is connected. Negative and positive signs suggest a two-way spectral partition.
3. Graph matrices
4. Spectrum and connected components
5. Walk counting
The theorem says that $(A^k)_{ij}$ counts walks of length $k$ from vertex $i$ to vertex $j$.
6. Laplacian energy
Use the signal $x_i=i$ on vertices. The tool compares $x^TLx$ with $\sum_{\{i,j\}\in E}(x_i-x_j)^2$.
7. Independent-study tasks with answers
Task A. Select the disconnected graph. How many zero Laplacian eigenvalues do you see?
Answer: The number of zero eigenvalues equals the number of connected components.
Answer: The number of zero eigenvalues equals the number of connected components.
Task B. Select two triangles + bridge. What partition is suggested by the Fiedler vector?
Answer: The two triangles should be separated into two clusters.
Answer: The two triangles should be separated into two clusters.
Task C. Select $P_6$. What does $(A^2)_{1,3}$ count?
Answer: The number of length-2 walks from vertex 1 to vertex 3.
Answer: The number of length-2 walks from vertex 1 to vertex 3.
Similar practice. Add or remove one bridge edge. How does the Fiedler value $\lambda_2$ change?
Answer: Stronger or more numerous connections between two parts usually increase $\lambda_2$.
Answer: Stronger or more numerous connections between two parts usually increase $\lambda_2$.