Brouwer degree for Kazdan-Warner equations on a connected finite graph

TL;DR

This paper investigates the solutions of Kazdan-Warner equations on finite graphs using degree theory, establishing bounds, computing the Brouwer degree, and providing new proofs for existing existence results.

Contribution

It introduces a degree-theoretic approach to analyze Kazdan-Warner equations on finite graphs, including solution bounds and Brouwer degree calculations.

Findings

Solutions are uniformly bounded for nonzero prescribed functions.

Brouwer degree is well defined and computed case by case.

New proofs of known existence results are provided.

Abstract

We study Kazdan-Warner equations on a connected finite graph via the method of the degree theory. Firstly, we prove that all solutions to the Kazdan-Warner equation with nonzero prescribed function are uniformly bounded and the Brouwer degree is well defined. Secondly, we compute the Brouwer degree case by case. As consequences, we give new proofs of some known existence results for the Kazdan-Warner equation on a connected finite graph.