Strong Isoperimetric Inequality for Tessellating Quantum Graphs

TL;DR

This paper establishes a strong isoperimetric inequality for tessellating quantum graphs by introducing a curvature-like measure and providing bounds and criteria for the isoperimetric constant's positivity.

Contribution

It introduces a novel curvature-like quantity for tessellating quantum graphs and proves a lower bound and positivity criterion for their isoperimetric constants.

Findings

Established a lower estimate for the isoperimetric constant.

Proved a criterion for the positivity of the isoperimetric constant.

Introduced a curvature-like measure for tessellating quantum graphs.

Abstract

We investigate isoperimetric constants of infinite tessellating metric graphs. We introduce a curvature-like quantity, which plays the role of a metric graph analogue of discrete curvature notions for combinatorial tessellating graphs. We then prove a lower estimate and a criterium for positivity of the isoperimetric constant.