----- > [!proposition] Proposition. ([[rows and columns of graph laplacian sum to zero]]) > The entries in a given row or column of the [[graph Laplacian]] $L$ for a [[graph]] $G$ sum to zero. > [!proof]- Proof. ([[rows and columns of graph laplacian sum to zero]]) > Letting $A$ denote the [[adjacency matrix]] of $L$ and fixing row $i$ of $L$, we compute $\sum_{j=1}^{n}G_{ij}=k_{i} + \sum_{j=1}^{n} A_{ij}=k_{i} + \sum_{j=1}^{n}-\mathbb{1}_{\{ j \text{ connects to } i \}}=k_{i} - k_{i}.$ > Thus row $i$ sums to $0$. Since $L$ is [[symmetric matrix|symmetric]], column $i$ does as well and we're done since $i$ was arbitrary. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```