Graph Laplacian and Signal Energy
Connect a graph’s matrix to signal differences, connected components and zero-energy patterns.
Free theory articleConnect a graph’s matrix to signal differences, connected components and zero-energy patterns.
Concept & interpretation
A graph can be represented by its Laplacian matrix L = D - A, where D is the degree matrix and A is the adjacency matrix. At vertex i, (Lx)i = Σj w_ij(xi-xj), the weighted sum of differences between its signal and each neighbor. The associated quadratic form computes the total energy of the signal across all edges.
Investigate Graph Connectivity and Zero Energy
Open Graph Laplacian and Signal Energy to experiment with different graph structures.
- Select the "Two pairs" graph and set the Edge weight to 2. Set the signals to [1, 1, -1, -1].
- Observe that the signal energy drops to 0, even though the signal vector is not all zeros.
- This happens because the graph has two disconnected components, allowing each component to hold a constant but different signal value without contributing to edge differences.
- Verify that the number of zero eigenvalues (the nullity of L) matches the number of positive-weight connected components.
- Change the Edge weight to 0 and observe that all edges become inactive, the graph splits into 4 components, and the Laplacian becomes a zero matrix with 4 zero eigenvalues.