For a given N-vertex similarity graph $ G=(V,A) $ the eigenvalues of the unrenormalized (graph) Laplacian may be denoted as


$$ 0= \mu\_0 \leq \mu\_1 \leq ... \leq \mu\_N $$
where the corresponding eigenvectors may be written as


$$ v\_0 , v\_1 , ... ,v\_N $$
In the case where $ 0= \mu\_0 = \mu\_1 < \mu\_2 $ I find conflicting definitions for the Fiedler vector in the literature , namely $ v\_1 $(eigenvector corresponding to the second lowest eigenvalue?) or $ v\_2 $ (eigenvector corresponding to the lowest non-zero eigenvalue). Which one should one choose as Fiedler vector?


In addition things get ambiguous when the lowest non-zero eigenvalue is degenerated as well. For example assume $ N = 5 $ and $ A\_{ij} = 1$ if $i \neq j $ and $0$ else. The eigenvalues of the Laplacian are (0,5,5,5,5) in this case. Which eigenvector corresponds to the Fiedler vector in this case?


A good reference would be appreciated !



exact definition of Fiedler vector
The concept of a Fiedler vector is defined for graphs that consist of one single connected component. Since the number of zero eigenvalues counts the number of connected components, the second largest eigenvalue $\nu\_1$ is then always nonzero. The multiplicity $m\_1$ of the second largest eigenvalue may be greater than one, in which case there is more than a single Fiedler vector.


One might ask why not extend the definition of the Fiedler vector to graphs with more than a single connected component. This is not particularly useful, as explained in this [MO posting](http://mathoverflow.net/questions/120336/can-the-first-non-zero-eigenvalue-of-a-laplacian-matrix-with-more-than-1-zero-va).


