Degree matrix

The degree matrix of a graph is diagonal, with each vertex’s degree on the corresponding diagonal entry. Its off-diagonal entries are zero.

Graph: A graph is a mathematical structure consisting of vertices joined by edges. The degree matrix encodes vertex information from this underlying structure.
Laplacian matrix: The graph Laplacian is the difference between a degree matrix and an adjacency matrix. It combines the degree matrix with adjacency to capture graph structure algebraically.
Spectral clustering: Spectral clustering partitions data using eigenvectors of a graph-derived matrix. Normalized Laplacians built with degree matrices help account for uneven cluster connectivity.
Weighted graph: A weighted graph assigns numerical weights to its edges. Its degree matrix may use weighted sums rather than counts of incident edges.
Vertex degree: The degree of a vertex is the number of incident edge ends, with conventions varying for loops. Each diagonal entry records the degree of one vertex.
Normalized graph Laplacian: A graph Laplacian scaled by vertex degrees, commonly as D⁻¹ᐟ²LD⁻¹ᐟ². Its normalization uses the degree matrix to adjust for differences in vertex degree.
Cheeger inequality: Cheeger inequalities relate graph expansion to eigenvalues of graph Laplacians. The degree matrix defines the volume terms and normalized operators in graph versions.
Directed graph: A directed graph has edges with specified orientations from one vertex to another. It distinguishes in-degree and out-degree, yielding different degree matrices.
Adjacency matrix: An adjacency matrix records which pairs of vertices in a graph are adjacent. Its row sums give vertex degrees for simple undirected graphs.
Random walk: A random walk is a process that moves between states according to transition probabilities. Dividing adjacency entries by vertex degrees produces transition probabilities on an undirected graph.

Linked from 4 pages

Kirchhoff's theoremRelated: Subtracting the adjacency matrix from it gives the combinatorial Laplacian.