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godin_thibault:brouillon_projet [2013/12/13 14:21]
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-            __Brouillon projet__+                                     ​Brouillon projet
  
  
-Laplacian Matrice+__Introduction__
  
 +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- $L=D-A ~\text{where }~ A ~\text{is the adjacency matrice and }~ D ~\text{the degree matrice.}$
 +
 +$L$ 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.1386940875.txt.gz · Last modified: 2013/12/13 14:21 by tgodin