Harnack Inequality for Magnetic Graphs

TL;DR

This paper establishes a Harnack inequality for eigenfunctions of the magnetic Laplace operator on magnetic graphs satisfying a curvature-dimension inequality, with applications to eigenvalue bounds and magnetic Cheeger numbers.

Contribution

It introduces a Harnack inequality for magnetic graphs under the $CD^\sigma(n,\kappa)$ condition, extending previous work to magnetic settings.

Findings

Derived a lower bound for the least eigenvalue based on curvature and graph parameters.

Established a relationship between the magnetic Cheeger number and graph curvature.

Provided new tools for analyzing spectral properties of magnetic graphs.

Abstract

For magnetic graphs satisfying connection curvature dimension inequality $CD^\sigma(n,\kappa)$, we prove a Harnack-type inequality for eigenfunctions of the graph magnetic Laplace operator in the manner of work done by Chung, Lin, Yau in 2014. Then we look at two applications; first a lower bound for the least eigenvalue in terms of curvature and extremal path/degree quantities, then to the magnetic Cheeger number of the graph.