Nodal domain theorem for the graph p-Laplacian.
Francesco Tudisco, Matthias Hein · arXiv (Cornell University) · 2016
In this work we consider the nonlinear graph $p$-Laplacian and the set of eigenvalues and associated eigenvectors of this operator defined by a variational principle. We prove a unifying nodal domain theorem for the graph $p$-Laplacian for any $p\geq 1$. While for $p>1$ the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian ($p=2$), the behavior changes for $p=1$. We show that the bounds are tight for $p\geq 1$ by studying the eigenvectors of the graph $p$-Laplacian for two example graphs where the bounds on the nodal domains are attained.