Topological Poincar\'e type inequalities and lower bounds on the infimum of the spectrum for graphs

TL;DR

This paper investigates topological Poincaré inequalities on graphs, characterizes graphs satisfying these inequalities, and relates the best constants to geometric diameters, inradii, and spectral gaps of Laplacians.

Contribution

It provides a comprehensive characterization of graphs with Poincaré inequalities and links the constants to geometric and spectral properties, offering new insights into graph analysis.

Findings

Characterization of graphs satisfying Poincaré inequalities

Geometric interpretation of constants as diameters and inradii

Variational formula relating constants to spectral gaps

Abstract

We study topological Poincar\'e type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory of Laplacians once a probability measure on the graph is chosen. More specifically, we obtain a variational characterization of these constants as infimum over spectral gaps of all Laplacians on the graphs associated to probability measures.