>[!info] Definition - Degree Matrix >**Diagonal** [[Matrices|matrix]] based on a given [[Graph|graph]], where each diagonal entry represents the [[Node Degree|degree]] of the corresponding node. > >For undirected graphs, it is defined based on the [[Adjacency Matrix|adjacency matrix]] via $\mathbf{D}= \text{diag}(\mathbf{A}\cdot \mathbb{1}_{n}) \in \mathbb{R}^{n\times n}.$In the directed case, we differentiate between >- **Out-Degree Matrix**$\mathbf{D}_{\text{out}}=\text{diag}(\mathbf{1^{T}A})$ >- **In-Degree Matrix**$\mathbf{D}_{\text{in}}=\text{diag}(\mathbf{A1})$ --- #### As a linear Operator The matrix-vector product $\mathbf{D}\cdot \mathbf{x}$ defines a **scaling** of a [[Graph Signal|graph signal's]] values at each [[Node Degree|node]] by its degree.