Dirichlet p-Laplacian eigenvalues and Cheeger constants on symmetric graphs

TL;DR

This paper investigates the eigenvalues and eigenfunctions of p-Laplacians with Dirichlet boundary conditions on symmetric graphs, linking eigenvalues to Cheeger constants and providing explicit calculations for specific graph classes.

Contribution

It characterizes the first eigenfunction of p-Laplacians on graphs and connects Cheeger constants of symmetric graphs to quotient graphs, offering new computational methods.

Findings

Characterization of the first eigenfunction via sign conditions.

Identification of Cheeger constants through eigenvalues as p approaches 1.

Explicit calculations of Cheeger constants for spherically symmetric graphs.

Abstract

In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.