Convex and subharmonic functions on graphs

TL;DR

This paper investigates the conditions under which convex functions on graphs are also subharmonic, focusing on lattice-like graphs generated by normed abelian groups, and establishes that all convex functions are subharmonic in this setting.

Contribution

It introduces a specific notion of convexity on lattice-like graphs and proves that convex functions are necessarily subharmonic within this class.

Findings

Convex functions are subharmonic on lattice-like graphs generated by normed abelian groups.

More structure is needed beyond basic convexity to ensure subharmonicity.

The paper establishes a clear relationship between convexity and subharmonicity in a specific graph setting.

Abstract

We explore the relationship between convex and subharmonic functions on discrete sets. Our principal concern is to determine the setting in which a convex function is necessarily subharmonic. We initially consider the primary notions of convexity on graphs and show that more structure is needed to establish the desired result. To that end, we consider a notion of convexity defined on lattice-like graphs generated by normed abelian groups. For this class of graphs, we are able to prove that all convex functions are subharmonic.