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godin_thibault:brouillon_projet [2013/12/13 15:14] tgodin |
godin_thibault:brouillon_projet [2013/12/13 16:20] (current) tgodin |
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- | Hence $L$ is diagonalisable and has it eigen values positive. In fact $0$ is always an eigenvalue, associated to the eigenvector ${}^t(1,\hdots,1)$ (it follows of the observation one). | + | Hence $L$ is diagonalisable and has it eigen values positive $\lambda_0, ..., \lambda_n$. In fact $0$ is always an eigenvalue, associated to the eigenvector ${}^t(1,...,1)$ (it follows of the observation one). |

One of the first application of this matrice has been the computation of the number of spanning trees, known as //Kirchnoff Matrice-Tree theorem// following and whose proved in appendix. | One of the first application of this matrice has been the computation of the number of spanning trees, known as //Kirchnoff Matrice-Tree theorem// following and whose proved in appendix. | ||

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__*Kirchnoff Matrice-Tree theorem__ The number of spanning trees is equal to any cofactor of the laplacian matrice. | __*Kirchnoff Matrice-Tree theorem__ The number of spanning trees is equal to any cofactor of the laplacian matrice. | ||

- | $\forall (i,j) \in \{1;n\}^2,~N_t(G)=\text{Det}\tilde{L}_{i,j}$ where $\tilde{L} \in M_{n-1}(\bf{Z})~ \tilde{L}_$ is the matrice $L$ where line $i$ and row $j$ have been deleted.. | + | $\forall (i,j) \in \{1;n\}^2,~N_t(G)=\text{Det}\tilde{L}_{i,j}$ where $\tilde{L}_{i,j} \in M_{n-1}(\bf{Z})~$ is the matrice $L$ where line $i$ and row $j$ have been deleted.. |

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In 1973, Miroslav Fiedler introduce in its article //Algebraic Connectivity in Graph// an interpretation of the second smallest eigen value of $L$ that we will denote $a(G)$ in term of connectivity. | In 1973, Miroslav Fiedler introduce in its article //Algebraic Connectivity in Graph// an interpretation of the second smallest eigen value of $L$ that we will denote $a(G)$ in term of connectivity. | ||

+ | First $(n-1)I-L$ is symetric, has its coefficients positives and admit $n-1~-\lambda_i$ as eigen value. As it is irreductible if $G$ is connected, by //Perron-Frobenius theorem// the greatest eigenvalue as multiplicity one. Otherwise if $G$ is not connected 0 as clearly an associated eigenspace of dimension greater than 2. Hence $a(G)\neq 0 \Leftrightarrow G~ \text{is connected}$ | ||

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+ | The eigenvector associated with $a(G)$, often called //Fielder vector// can be use as a mesure of the connectivity of a given vertice. | ||

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+ | __Computation of $a(G)$__ | ||

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+ | //Courant theorem// can be used to compute $a(G)$ : | ||

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+ | \[a(G) = \text{min}_{||x||=1}~({}^txLx) | ||

+ | \] | ||

__Bibliographie__ | __Bibliographie__ |

/net/html/perso/melancon/Visual_Analytics_Course/data/attic/godin_thibault/brouillon_projet.1386944064.txt.gz · Last modified: 2013/12/13 15:14 by tgodin