{{Short description|Type of matrix in algebraic graph theory}} 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.<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. |editor-link=L. W. Beineke | editor2-last = Wilson | editor2-first = Robin J. |editor2-link=Robin Wilson (mathematician) | 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 |- |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 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)