In the mathematical field of algebraic graph theory, the degree matrix of an undirected graph is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edges attached to each vertex.[1] It is used together with the adjacency matrix to construct the Laplacian matrix of a graph: the Laplacian matrix is the difference of the degree matrix and the adjacency matrix.[2]

## 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[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).

## Example

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

Vertex labeled graph Degree matrix
${\displaystyle {\begin{pmatrix}4&0&0&0&0&0\\0&3&0&0&0&0\\0&0&2&0&0&0\\0&0&0&3&0&0\\0&0&0&0&3&0\\0&0&0&0&0&1\\\end{pmatrix))}$

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).

## Properties

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.

## References

1. ^ a b Chung, Fan; Lu, Linyuan; Vu, Van (2003), "Spectra of random graphs with given expected degrees", Proceedings of the National Academy of Sciences of the United States of America, 100 (11): 6313–6318, Bibcode:2003PNAS..100.6313C, doi:10.1073/pnas.0937490100, MR 1982145, PMC 164443, PMID 12743375.
2. ^ Mohar, Bojan (2004), "Graph Laplacians", in Beineke, Lowell W.; Wilson, Robin J. (eds.), Topics in algebraic graph theory, Encyclopedia of Mathematics and its Applications, vol. 102, Cambridge University Press, Cambridge, pp. 113–136, ISBN 0-521-80197-4, MR 2125091.