Degree matrix - Revision history

Vipul at 19:26, 25 May 2014

2014-05-25T19:26:00Z

← Older revision		Revision as of 19:26, 25 May 2014
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	* For <math>1 \le i \le n</math> the <math>(i,i)^{th}</math> (diagonal) entry of the matrix is the [[degree of a vertex\|degree]] of the vertex <math>v(i)</math>.		* For <math>1 \le i \le n</math> the <math>(i,i)^{th}</math> (diagonal) entry of the matrix is the [[degree of a vertex\|degree]] of the vertex <math>v(i)</math>.

	Note that ~~the degree~~ matrix is ~~sensitive to~~ the ~~ordering of~~ the ~~vertices chosen. Permuting~~ the ~~ordering~~ of ~~the vertices affects~~ the degree matrix ~~through conjugation~~ by ~~that~~ permutation matrix. In particular, the characteristic polynomial as well as the multiset of eigenvalues for the degree matrix does not depend on the ordering of vertices chosen.		Note that this matrix is well defined only ''after'' we fix the bijection <math>v</math>. Changing the bijection has the effect of [[groupprops:conjugation\|conjugating]] the degree matrix by a permutation matrix obtained by composing one bijection with the inverse of the other. In other words, we re-label the rows and re-label the columns in the same manner. In particular, the characteristic polynomial as well as the multiset of eigenvalues for the degree matrix does not depend on the ordering of vertices chosen.

	==Related notions==		==Related notions==

	* The [[degree sequence of a graph]] is informationally equivalent to the degree matrix. The former stores the degrees as a non-increasing sequence of degrees, while the latter uses a matrix structure for storing the information.		* The [[degree sequence of a graph]] is informationally equivalent to the degree matrix. The former stores the degrees as a non-increasing sequence of degrees, while the latter uses a matrix structure for storing the information.

Vipul at 19:16, 25 May 2014

2014-05-25T19:16:12Z

← Older revision		Revision as of 19:16, 25 May 2014
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	The '''degree matrix''' of a graph is a [[linear:diagonal matrix\|diagonal matrix]] where the rows and columns are indexed by the set of vertices (in the same order), and each diagonal entry gives the [[degree of a vertex\|degree]] of the corresponding vertex.		The '''degree matrix''' of a graph is a [[linear:diagonal matrix\|diagonal matrix]] where the rows and columns are indexed by the set of vertices (in the same order), and each diagonal entry gives the [[degree of a vertex\|degree]] of the corresponding vertex.

			In words:

			Suppose <math>G</math> is a [[finite graph\|finite]] [[undirected graph]]. Let <math>n</math> be the size of the vertex set <math>V(G)</math>. Fix a bijective correspondence <math>v:\{ 1,2,\dots,n\} \to V(G)</math>. The '''degree matrix''' of <math>G</math> is a <math>n \times n</math> diagonal matrix such that:

			* For <math>i \ne j</math> with <math>1 \le i,j \le n</math>, the <math>(i,j)^{th}</math> entry of the matrix is 0.
			* For <math>1 \le i \le n</math> the <math>(i,i)^{th}</math> (diagonal) entry of the matrix is the [[degree of a vertex\|degree]] of the vertex <math>v(i)</math>.

	Note that the degree matrix is sensitive to the ordering of the vertices chosen. Permuting the ordering of the vertices affects the degree matrix through conjugation by that permutation matrix. In particular, the characteristic polynomial as well as the multiset of eigenvalues for the degree matrix does not depend on the ordering of vertices chosen.		Note that the degree matrix is sensitive to the ordering of the vertices chosen. Permuting the ordering of the vertices affects the degree matrix through conjugation by that permutation matrix. In particular, the characteristic polynomial as well as the multiset of eigenvalues for the degree matrix does not depend on the ordering of vertices chosen.

Vipul: Created page with "==Definition== The '''degree matrix''' of a graph is a diagonal matrix where the rows and columns are indexed by the set of vertices (in the same o..."

2014-05-25T19:11:05Z

Created page with "==Definition== The '''degree matrix''' of a graph is a diagonal matrix where the rows and columns are indexed by the set of vertices (in the same o..."

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==Definition==

The '''degree matrix''' of a graph is a [[linear:diagonal matrix|diagonal matrix]] where the rows and columns are indexed by the set of vertices (in the same order), and each diagonal entry gives the [[degree of a vertex|degree]] of the corresponding vertex.

Note that the degree matrix is sensitive to the ordering of the vertices chosen. Permuting the ordering of the vertices affects the degree matrix through conjugation by that permutation matrix. In particular, the characteristic polynomial as well as the multiset of eigenvalues for the degree matrix does not depend on the ordering of vertices chosen.

==Related notions==

* The [[degree sequence of a graph]] is informationally equivalent to the degree matrix. The former stores the degrees as a non-increasing sequence of degrees, while the latter uses a matrix structure for storing the information.