# What do the eigenvectors of an adjacency matrix tell us?

The principal eigenvector of the adjacency matrix of a graph gives us some notion of vertex centrality.

What do the second, third, etc. eigenvectors tell us?

Motivation: A standard information retrieval technique (LSI) uses a truncated SVD as a low-rank approximation of a matrix. If we truncate to rank 1, then we essentially have a PageRank algorithm. I was wondering if there are ways of interpreting the corrections introduced by higher eigenvectors.

Something similar to what the moments of a distribution tell us (e.g. first moment gives us the mean, second tells us the variance, third gives us skewness, etc).

• The second smallest eigenvector can be used to partition the graph into clusters, via spectral clustering from wikipedia. It's would be interesting to see the usage of all different eigenvectors
– com
Mar 6, 2012 at 10:15
• My note ---"Spectral Realizations of Graphs" ( daylateanddollarshort.com/math/pdfs/spectral.pdf )--- explains that a matrix whose rows form an orthogonal basis of an eigenspace of a graph's adjacency matrix has columns that serve as the coordinate vectors of a geometric realization for which automorphisms are induced by rigid isometries. That is, eigenvectors construct visual models of a graph's automorphic structure. The exact contribution a particular eigen-model makes to the understanding of that structure isn't necessarily obvious, but the pictures can be pretty cool. :)
– Blue
Mar 9, 2012 at 0:51
• Why do you ask about the spectrum of the adjacency as opposed to that of the signed and/or unsigned graph Laplacian matrix, which are the typical objects of study in spectral graph theory? Eg, math.ucsd.edu/~fan/research/cb/ch1.pdf, buzzard.ups.edu/bookreview/eigenspaces-graphs-beezer-review.pdf Jul 17, 2012 at 1:17