Editing Degree matrix

{{Short description|Type of matrix in algebraic graph theory}}
In the [[mathematics|mathematical]] field of [[algebraic graph theory]], the '''degree matrix''' of an [[undirected graph]] is a [[diagonal matrix]] which contains information about the [[degree (graph theory)|degree]] of each [[vertex (graph theory)|vertex]]—that is, the number of edges attached to each vertex.<ref name="clv">{{citation
 | last1 = Chung | first1 = Fan | author1-link = Fan Chung
 | last2 = Lu | first2 = Linyuan
 | last3 = Vu | first3 = Van | author3-link = Van Vu
 | doi = 10.1073/pnas.0937490100
 | issue = 11
 | journal = Proceedings of the National Academy of Sciences of the United States of America
 | mr = 1982145
 | pages = 6313–6318
 | title = Spectra of random graphs with given expected degrees
 | volume = 100
 | year = 2003 | pmid=12743375 | pmc=164443| bibcode = 2003PNAS..100.6313C | doi-access = free }}.</ref> 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.<ref name="mohar">{{citation
 | last = Mohar | first = Bojan | authorlink = Bojan Mohar
 | editor1-last = Beineke | editor1-first = Lowell W.
 | editor2-last = Wilson | editor2-first = Robin J.
 | contribution = Graph Laplacians
 | isbn = 0-521-80197-4
 | mr = 2125091
 | pages = 113–136
 | publisher = Cambridge University Press, Cambridge
 | series = Encyclopedia of Mathematics and its Applications
 | title = Topics in algebraic graph theory
 | url = https://books.google.com/books?id=z2K26gZLC1MC&pg=PA113
 | volume = 102
 | year = 2004}}.</ref>

==Definition==
Given a graph <math>G=(V,E)</math> with <math>|V|=n</math>, the '''degree matrix''' <math>D</math> for <math>G</math> is a <math>n \times n</math> [[diagonal matrix]] defined as<ref name="clv"/>
:<math>D_{i,j}:=\left\{
\begin{matrix} 
\deg(v_i) & \mbox{if}\ i = j \\
0 & \mbox{otherwise}
\end{matrix}
\right.
</math>

where the degree <math>\deg(v_i)</math> 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:
 
{|class="wikitable"
![[Vertex labeled graph]]
!Degree matrix
|-
|[[Image:6n-graph2.svg|175px]]
|<math>\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}</math>
|}

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 <math>k</math>.

According to the [[degree sum formula]], the [[Trace (linear algebra)|trace]] of the degree matrix is twice the number of edges of the considered graph.

==References==
{{reflist}}

{{Matrix classes}}

[[Category:Algebraic graph theory]]
[[Category:Matrices (mathematics)]]