Degree matrix

Given a graph ${\displaystyle G=(V,E)}$ with ${\displaystyle |V|=n}$, the degree matrix ${\displaystyle D}$ for ${\displaystyle G}$ is a ${\displaystyle n\times n}$ diagonal matrix defined as[1]

${\displaystyle D_{i,j}:=\left\{{\begin{matrix}\deg(v_{i})&{\mbox{if}}\ i=j\\0&{\mbox{otherwise}}\end{matrix}}\right.}$

where the degree ${\displaystyle \deg(v_{i})}$ of a vertex counts the number of times an edge terminates at that vertex. In an undirected graph, this means that each loop increases the degree of a vertex by two. In a directed graph, the term degree may refer either to indegree (the number of incoming edges at each vertex) or outdegree (the number of outgoing edges at each vertex).

The degree matrix of a k-regular graph has a constant diagonal of ${\displaystyle k}$.