exact definition of Fiedler vector

Question

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 !

Answer 1

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.