Laplacian

Laplace operator

$ \Delta f:=(\nabla\cdot\nabla)f=\sum_{i=1}^n\frac{\partial^2 f}{\partial x_i^2}

橢圓型作用素 (elliptic operator)

Laplacian 行列 (Laplacian matrix。admittance 行列。Kirchhoff 行列。離散 Laplacian)

$ L_{ij}:=D-A=\begin{cases} {\rm deg}(v_i) & {\rm if}~i=j \\ -1 & {\rm if}~i\ne j~{\rm and}~v_i~{\rm is~adjacent~to}~v_j \\ 0 & {\rm otherwise} \end{cases}

$ L=D-A

$ L=BB^\top

$ A_{ij}:=\begin{cases} {\rm deg}(v_i) & {\rm if}~i=j \\ 0 & {\rm otherwise} \end{cases}