Kazdan-Warner equation on graph in the negative case

Let G=(V,E) be a connected finite graph. In this short paper, we reinvestigate the Kazdan-Warner equationΔu=c−heu with c<0 on G, where h defined on V is a known function. Grigor'yan, Lin and Yang [3] showed that if the Kazdan-Warner equation has a solution, then h‾, the average value of h, is negative. Conversely, if h‾<0, then there exists a number c−(h)<0, such that the Kazdan-Warner equation is solvable for every 0>c>c−(h) and it is not solvable for c