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 following undirected graph has a 6x6 degree matrix with values:

Note that in the case of undirected graphs, an edge that starts and ends in the same node increases the corresponding degree value by 2 (i.e. it is counted twice).

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

According to the degree sum formula, the trace of the degree matrix is twice the number of edges of the considered graph.