Degree matrix

Definition

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

${\displaystyle D_{i,j}:=\{\begin{array}{cc}\mathrm{deg}(v_i)& \text{if}i=j\\ 0& \text{otherwise}\end{array}}$

where the degree ${\displaystyle \mathrm{deg}(v_i)}$ 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).