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godin_thibault:brouillon_projet [2013/12/13 14:24]
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godin_thibault:brouillon_projet [2013/12/13 16:20] (current)
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-Laplacian Matrice +__Introduction__ 
-<​math>​L=D-A \text{where} A \text{is the adjacency matrice and } D \text{the degree matrice.}</math>+ 
 +Let $G=(V,E)$ be a simple graph of $n$ nodes. 
 +we define the laplacian matrice (also known as Kirchnoff matrice) ​ $L=(l_{i,​j})_{(i,​j) \in \{1;n^2\}}$ by 
 +\[l_{i,​j}:​=\left\{ 
 +\begin{matrix} 
 +-1 & \mbox{if }~ v_i~ \mbox{is adjacent to}~ v_j \\ 
 + 
 +\text{deg}(v_i) & \mbox{if }~ i=j\\ 
 +\end{matrix} 
 +\right. 
 +\] 
 + 
 +observation 1- $L=D-A ~\text{where }~\text{is the adjacency matrice and }~\text{the degree matrice.}
 + 
 +$L$ is a symetric, positive matrice. Indeed $L=U{}^tU$ where $U$ is an incidence matrice  
 +\[u_{i,​j}:​=\left\{ 
 +\begin{matrix} 
 +1 & \mbox{if} ~v_i \mbox{ is the target of } e_j \\ 
 +-1 & \mbox{if }~ v_i \mbox{ is the source of } e_j \\ 
 +0 & \mbox{if }~ v_i \notin e_j 
 +\end{matrix} 
 +\right. 
 +\] 
 +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. 
 + 
 + 
 +__*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 $\tilde{L}_{i,​j} \in M_{n-1}(\bf{Z})~$ is the matrice $L$ where line $i$ and row $j$ have been deleted.. 
 + 
 + 
 +__Connectivity Number__ 
 + 
 +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__ 
 + 
 +*M Fiedler, //Algebraic Connectivity in Graph//, Czechoslovak Mathematical Journal, 23 (98), 1973 
 + 
 +__Appendix__
/net/html/perso/melancon/Visual_Analytics_Course/data/attic/godin_thibault/brouillon_projet.1386941096.txt.gz · Last modified: 2013/12/13 14:24 by tgodin