Laplacian matrix of a graph

Let $G$ be a finite graph with $n$ vectices and let $D$ be the incidence matrix (http://planetmath.org/IncidenceMatrixWithRespectToAnOrientation) of $G$ with respect to some orientation.The Laplacian matrix of $G$ is defined to be $DD^{T}$ .

If we let $A$ be the adjacency matrix of $G$ then it can be shownthat $DD^{T}=\\Delta-A$ ,where $\\Delta=\\textrm{diag}(\\delta_{1},\\ldots,\\delta_{n})$ and $\\delta_{i}$ is the degree ofthe vertex $v_{i}$ . As a result, the Laplacian matrix is independent of what orientation ischosen for $G$ .

The Laplacian matrix is usually denoted by $L(G)$ . It is a positive semidefinitesingular matrix, so that the smallest eigenvalue is 0.

单词	LaplacianMatrixOfAGraph
释义	Laplacian matrix of a graph Let $G$ be a finite graph with $n$ vectices and let $D$ be the incidence matrix (http://planetmath.org/IncidenceMatrixWithRespectToAnOrientation) of $G$ with respect to some orientation.The Laplacian matrix of $G$ is defined to be $DD^{T}$ . If we let $A$ be the adjacency matrix of $G$ then it can be shownthat $DD^{T}=\\Delta-A$ ,where $\\Delta=\\textrm{diag}(\\delta_{1},\\ldots,\\delta_{n})$ and $\\delta_{i}$ is the degree ofthe vertex $v_{i}$ . As a result, the Laplacian matrix is independent of what orientation ischosen for $G$ . The Laplacian matrix is usually denoted by $L(G)$ . It is a positive semidefinitesingular matrix, so that the smallest eigenvalue is 0.
随便看	change of basis ChangeOfVariableInDefiniteIntegral change of variable in definite integral ChangeOfVariablesInIntegralOnmathbbRn change of variables in integral on ℝ^n ChaoticDynamicalSystem chaotic dynamical system Chapter10Notes Chapter 10 Notes Chapter11Notes Chapter 11 Notes Chapter1Notes Chapter 1 notes Chapter2Notes Chapter 2 notes Chapter3Notes Chapter 3 Notes Chapter4Notes Chapter 4 Notes Chapter5Notes Chapter 5 Notes Chapter6Notes Chapter 6 Notes Chapter7Notes Chapter 7 Notes