Graph Laplacian ($L$)

For a finite graph with adjacency matrix $A$ and degree matrix $D$, its graph Laplacian isIts quadratic form is $x^{T}Lx={\sum }_{\{i,j\}\in E}(x_i-x_j{)}^2$.

For symmetric edge weights $w_{ij}\ge 0$, the weighted graph Laplacian has quadratic form

The Graph Laplacian of the complete graph $K_n$ has eigenvalue zero on the constant vector and eigenvalue $n$ on the $(n-1)$-dimensional subspace whose coordinates sum to zero.