Finiteness of the graph p-Laplacian spectrum

Determine whether the spectrum of the graph p-Laplacian is always finite on general finite graphs. If it is finite, derive explicit upper bounds for the number of eigenvalues in terms of graph size and p.

Background

In contrast to the linear (p=2) Laplacian, where the spectrum is well-understood, the nonlinear p-Laplacian may have more eigenvalues than the space dimension and examples show rich behavior. Although structured families such as complete graphs and trees exhibit finite spectra, there is no general proof for arbitrary graphs. The authors also note that continuity and isolation properties of eigenvalues hinge on finiteness questions.

Because the p-Laplacian spectrum is bounded, the existence of isolated eigenvalues—and their continuity in p—is closely tied to whether the overall spectrum is finite. Establishing finiteness (and bounding the number of eigenvalues) would clarify multiplicity notions and spectral regularity.