Sobolev-type inequalities and eigenvalue growth on graphs with finite measure

Abstract

In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds can not be established.

Universal lower bounds for Laplacians on weighted graphs

“…In this section we put our results in the context of Poincaré inequalities on graphs, see [17,12] for related recent results as well. This will be used to discuss optimality.…”

Existence and multiplicity of solutions to p-Laplacian equations on graphs

“…For example, Lin and Wu [16] studied the existence and nonexistence of global solutions for a semilinear heat equation on graphs. Hua, Keller, Schwarz and Wirth [12] investigated the eigenvalue growth of infinite graphs with discrete spectrum. Cushing, Kamtue, Liu and Peyerimhoff [5] reformulated the Bakry-Émery curvature on a weighted graph.…”