reduced Laplacian matrix

Let $G$ be a graph with $n$ vertices. Fix some $i$ , a vertex. The reduced Laplacian matrix with respect to $i$ is ${\tilde{L}}_{i}$ , obtained by removing the $i$ th row and column from $L$ the Laplacian (Kirchoff) matrix.

Example

Let $G$ be the following graph:
Pasted image 20260409095759.png|200
Its $i = 4$ reduced Laplacian matrix is

{\tilde{L}}_{4} = [\begin{matrix} 4 & - 1 & - 1 \\ - 1 & 2 & 0 \\ - 1 & 0 & 2 \end{matrix}]