Degree Matrix

The degree matrix of a graph is a square matrix that records how many edges meet at each vertex. For an undirected graph with n vertices, the degree matrix is an n by n diagonal matrix whose diagonal entries give each vertex degree, the number of edges attached to it, while every entry off the diagonal is zero; a self-loop at a vertex counts twice toward that vertex degree. Subtracting the adjacency matrix from the degree matrix produces the graph Laplacian matrix, a tool used throughout spectral graph theory. In a k-regular graph, where every vertex has the same degree, the degree matrix has the same constant value running down its whole diagonal.