Eigenvalues of the Discrete \(p\)-Laplacian for Graphs

Abstract

We study the discrete version of the \(p\)-Laplacian operator — \(\textrm{div}(|\nabla u|^{p-2}\nabla u)\) — and we give some estimates of its smallest positive eigenvalue. In earlier papers, eigenvalues of the discrete Laplacian have been considered. We shall here study more general means. We shall also, in particular, study the case when the graph is complete. We give an estimate of the smallest positive eigenvalue of the \(p\)-Laplacian when the graph is a subgraph of \(\mathbb{Z}^n\) in this context. We give all eigenvalues of the \(p\)-Laplacian when the graph is complete.