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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 Both sides previous revision Previous revision 2013/12/13 16:20 tgodin 2013/12/13 15:52 tgodin 2013/12/13 15:39 tgodin 2013/12/13 15:15 tgodin 2013/12/13 15:14 tgodin 2013/12/13 14:52 tgodin 2013/12/13 14:50 tgodin 2013/12/13 14:27 tgodin 2013/12/13 14:27 tgodin 2013/12/13 14:24 tgodin 2013/12/13 14:21 tgodin 2013/12/13 14:21 tgodin created Next revision Previous revision 2013/12/13 16:20 tgodin 2013/12/13 15:52 tgodin 2013/12/13 15:39 tgodin 2013/12/13 15:15 tgodin 2013/12/13 15:14 tgodin 2013/12/13 14:52 tgodin 2013/12/13 14:50 tgodin 2013/12/13 14:27 tgodin 2013/12/13 14:27 tgodin 2013/12/13 14:24 tgodin 2013/12/13 14:21 tgodin 2013/12/13 14:21 tgodin created Line 26: Line 26: \right. \right. \] \] - 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. Line 32: Line 32: __*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.. Line 39: Line 39: 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 $a(G)\neq 0 \Leftrightarrow G~ \text{is connected}$ + + The eigenvector associated with $a(G)$, often called //Fielder vector// can be use as a mesure of the connectivity of a given vertice. + + + + __Computation of $a(G)$__ + + + //Courant theorem// can be used to compute $a(G)$ : + + $a(G) = \text{min}_{||x||=1}~({}^txLx) +$ __Bibliographie__ __Bibliographie__