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godin_thibault:brouillon_projet [2013/12/13 14:52]
tgodin
godin_thibault:brouillon_projet [2013/12/13 16:20] (current)
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                                      ​Brouillon projet                                      ​Brouillon projet
 +
 +
 +__Introduction__
  
 Let $G=(V,E)$ be a simple graph of $n$ nodes. Let $G=(V,E)$ be a simple graph of $n$ nodes.
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 \] \]
  
-$L=D-A ~\text{where }~ A ~\text{is the adjacency matrice and }~ D ~\text{the degree matrice.}$+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 ​ $L$ is a symetric, positive matrice. Indeed $L=U{}^tU$ where $U$ is an incidence matrice ​
 \[u_{i,​j}:​=\left\{ \[u_{i,​j}:​=\left\{
 \begin{matrix} \begin{matrix}
-1 & \mbox{if} v_i \mbox{ is the target of } e_j \\ +1 & \mbox{if} ​~v_i \mbox{ is the target of } e_j \\ 
--1 & \mbox{si } v_i \mbox{ is the source of } e_j \\ +-1 & \mbox{if }v_i \mbox{ is the source of } e_j \\ 
-0 & \mbox{si } v_i \notin e_j+0 & \mbox{if }v_i \notin e_j
 \end{matrix} \end{matrix}
 \right. \right.
 \] \]
-Hence $L$ is diagonalisable ​+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.1386942757.txt.gz · Last modified: 2013/12/13 14:52 by tgodin