  We investigate the first and second moments of the inverse participation
ratio (IPR) for all eigenvectors of the Laplacian on finite random regular
graphs with $n$ vertices and degree $z$. By exactly diagonalizing a large set
of $z$-regular graphs, we find that as $n$ becomes large, the mean of the
inverse participation ratio on each graph, when averaged over a large ensemble
of graphs, approaches the numerical value $3$. This universal number is
understood as the large-$n$ limit of the average of the quartic polynomial
corresponding to the IPR over an appropriate $(n-2)$-dimensional hypersphere of
$\mathbb{R}^n$. For a large, but not exhaustive ensemble of graphs, the mean
variance of the inverse participation ratio for all graph Laplacian
eigenvectors deviates from its continuous hypersphere average due to large
graph-to-graph fluctuations that arise from the existence of highly localized
modes.
