Topological degree for negative fractional Kazdan--Warner equation on finite graphs

TL;DR

This paper explores the fractional Kazdan--Warner equation on finite graphs using topological degree theory, establishing existence and multiplicity results and extending prior work to the fractional case.

Contribution

It extends existing results to the fractional setting and provides a concise proof for related equations on finite graphs.

Findings

Established existence of solutions for the fractional Kazdan--Warner equation

Proved multiplicity of solutions under certain conditions

Extended previous results to fractional equations on graphs

Abstract

Studies on Kazdan--Warner equations on graphs have grown steadily, yet the fractional case remains insufficiently explored. Using topological degree theory, this work investigates the fractional Kazdan--Warner equation in the negative case on connected finite graphs, focusing on the existence and multiplicity of solutions. This work not only extends the earlier result of S. Liu and Yang (2020) to the fractional setting, but also provides a concise proof for the work of Shan and Y. Liu (2025).