  We address the problem of computing the graph $p$-Laplacian eigenpairs for
$p\in (2,\infty)$. We propose a reformulation of the graph $p$-Laplacian
eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue
problem and discuss theoretical and computational advantages. We provide a
correspondence between $p$-Laplacian eigenpairs and linear eigenpair of a
constrained generalized weighted Laplacian eigenvalue problem. As a result, we
can assign an index to any $p$-Laplacian eigenpair that matches the Morse index
of the $p$-Rayleigh quotient evaluated at the eigenfunction. In the second part
of the paper we introduce a class of spectral energy functions that depend on
edge and node weights. We prove that differentiable saddle points of the $k$-th
energy function correspond to $p$-Laplacian eigenpairs having index equal to
$k$. Moreover, the first energy function is proved to possess a unique saddle
point which corresponds to the unique first $p$-Laplacian eigenpair. Finally we
develop novel gradient-based numerical methods suited to compute $p$-Laplacian
eigenpairs for any $p\in(2,\infty)$ and present some experiments.
