# Is there a simple interpretation of the eigenvectors of a graph (visualizable?)?

I want to understand eigenvectors obtain from graphs (adjacency matrices) in an analogous way as they are interpreted from principal component analysis of a set of images, which is easy:Eigenfaces Wiki. From the wiki page we see that each row in the matrix is an image and the eigenvector a reference image:

Each image is treated as one vector, simply by concatenating the rows of pixels in the original image, resulting in a single row with r × c elements. For this implementation, it is assumed that all images of the training set are stored in a single matrix T, where each row of the matrix is an image. Subtract the mean. The average image a has to be calculated and then subtracted from each original image in T. Calculate the eigenvectors and eigenvalues of the covariance matrix S. Each eigenvector has the same dimensionality (number of components) as the original images, and thus can itself be seen as an image. The eigenvectors of this covariance matrix are therefore called eigenfaces.

Can we then say that the adjacency matrix is a set of row networks with outgoing edges, and that the eigenvector is another network as well? I am looking for a tangible interpretation of eigenvector of the graph's adjacency matrix in a similar fashion of eigenfaces. (something that could also be visualized easily?)

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The adjacency matrix of a graph is square (= number of graph vertices) and will generally have as many eigenvectors as the dimensions as the adjacency (though not necessarily linearly independent), not just one. - But yes in principle you can map eigenspaces to gray scale values and render visually just like Eigenfaces. – alancalvitti Jan 31 '13 at 15:41