Cardinal utility - Revision history

en>John of Reading: /* Expected utility theory */Typo fixing, replaced: modifiying → modifying using AWB

2015-01-10T18:45:34Z

Expected utility theory: Typo fixing, replaced: modifiying → modifying using AWB

59.90.101.46: /* Analytic properties */

2013-11-11T10:03:06Z

Analytic properties

← Older revision		Revision as of 12:03, 11 November 2013
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			In the [[mathematics\|mathematical]] field of [[graph theory]] the '''degree matrix''' 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. It is used together with the [[adjacency matrix]] to construct the [[Laplacian matrix]] of a graph.

			==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
			:<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 new 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==
			{\|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>
			\|}



			*The degree matrix of a [[k-regular graph]] has a constant diagonal of <math>k</math>

			{{DEFAULTSORT:Degree Matrix}}
			[[Category:Algebraic graph theory]]
			[[Category:Matrices]]

en>Forich: /* Expected utility theory */ formatting

2012-08-25T22:37:46Z

Expected utility theory: formatting

New page

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