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

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).