Lipschitz harmonic functions on vertex-transitive graphs

TL;DR

This paper proves that all locally finite vertex-transitive graphs support non-constant Lipschitz harmonic functions, revealing a fundamental property of such symmetric graphs.

Contribution

It establishes the existence of non-constant Lipschitz harmonic functions on all locally finite vertex-transitive graphs, a novel theoretical result.

Findings

Existence of non-constant Lipschitz harmonic functions on these graphs

Extension of harmonic analysis to symmetric graph structures

New insights into the structure of vertex-transitive graphs

Abstract

We prove that every locally finite vertex-transitive graph $G$ admits a non-constant Lipschitz harmonic function.