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godin_thibault:brouillon_projet [2013/12/13 15:39]
tgodin
godin_thibault:brouillon_projet [2013/12/13 16:20] (current)
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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 $a(G)\neq 0 \Leftrightarrow G \text{is connected}$+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__
  
/net/html/perso/melancon/Visual_Analytics_Course/data/attic/godin_thibault/brouillon_projet.1386945598.txt.gz · Last modified: 2013/12/13 15:39 by tgodin